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———— 


PRESENTED BY 
PROF. E. G. LEWIS 
FROM THE LIBRARY OF 
Mime) FeO oem he 





Wy 
MAFHEMARGS 
LIBRARY, 


Digitized by the Internet Archive 
in 2022 with funding from 
University of Illinois Urbana-Champaign 


https://archive.org/details/courseofmathematO2hutt_0O 





COURSE 


MATHEMATICS 


FOR THE ‘eg 


USE OF ACADEMIES, 


AS WELL AS 


PRIVATE TUITION. 





IN TWO VOLUMES. 


BY 


CHARLES HUTTON, LL.D. F.R.5. 


Late Professor of Mathematics in the Royal Military Academy. 


THE THIRD AMERICAN EDITION: 
From the Fifth, Sixth, and Seventh London Editions. 


REVISED, CORRECTED, AND IMPROVED. 


TO WHICH iS ADDED 
An Elementary Essay on Descriptive Geometry. 
By ROBERT ADRAIN, LL. D. F.A. P.S. F.A.A.S, &c. 


And Professor of Mathematics and Natural Philosophy, in Columbia 
College, New-York. 


VOL. II. 


NEW-YORK: 
PUBLISHED BY SAMUEL CAMPBELL & SON, EVERT DUYCKINCK, T. & 5. sworns, 


R, M’DERMUT, T. A. RONALDS, COLLINS & HANNAY, AND GEORGE LONG. 
George Long, Printer, 71, Pear!l-Street.. 


1822. 


es 
epee ts ae 1 abies Remini 3 iy 


= 


: OES. She's ara 
yon. } 3 eka Coe aly A 
mee awed A nest 





722 CONTENTS 
OF VOLUME II. 


—_—— 


PLANE TRIGONOMETRY considered analytically ome 


Spherical Trigonometry . ; : ; : ‘ 

On Geodesic Operations, and the Figure of the Earth 

Principles of Polygonometry . . .« + © «© ¢ 

‘ Of Motion, Forces . 4 : : : Sa vase goer fe 
General Lawsof Motion. . . . . A ele Pelee haa 


Collision of Bodies . ae calls ‘rata 


Laws of Gravity—Descent of ham bodies—Motion of yA al in 
Sree space. : : 


Practical Gunnery . K : : i , : . 


Descent of bodies on inclined planes and curve surfaces—Motion of 


pendulums . : : ‘. : : thats “ ; ¥ 
The Mechanical Dascars SAUTE oP ES Taq MMD htt ute sake!) MT alk Ai can 
Centre of Gravity . : w Pip lores ite 4 . ane each 
Strength and stress of beams, &c. 


Centre of Percussion : i : 
Centre of Oscillation : shes ths ; ‘ . . ; i 
Centre of Gyration . ites : f ; . Whe ollie 


Of Hydrostatics sy : Pe * RRs matali 

Of Hydraulics suns 

Of Pneumatics : ; aad algae ces ney a Fite 

Of the Syphon . ; i : : ; 

Of the AirePUmnp ooo 6 ne ne ee 

Diving-bell and condensing machine 

Barometer : : : 3 ‘ ; : : : 

Thermometer . 3 A 3 5 oo). de ; * 

On the resistance of Fluids, and with thar forces and actions on bodies 

Practical Exercises concerning SpecificGravity . . . .. 

Of the Piling of Balls and Shells 7 

Of Distances by the velocity of Sound . : : 

Practical Exercises in Mechanics, Statics, Hydraulics, Sound, Motion, 
Gravity, Projectiles, end other branches of Natural Philosophy 


iv | CONTENTS. 


On the Nature and Solution of Equations in gencral 


On. the Nature and Properties of Curves, and the Construction of Geo- 


metrical Problems 


THE DOCTRINE OF FLUXIONS. 


Definitions and Principles  . . 
Direct Method of Fluxions 

Of second, third, &c. Fluxions | 

The Inverse Method, or The Finding of F Fluents 


Of Fluxions and Fluents 3 : ; . , i 


Of Maxima and Minima 

The Method of Tangents : 
Of Rectifications, or, to find the lengths of Curve Lines . 
of Quadratures ‘ A : ita : i 
To Sind the Surfaces of Solids . 

To find the Contents of Solids . } efits 

To find Logarithms 

To find the points of Inflection 4 

To find the radius of curvature of Curves. 

Of Involute and Evolute Curves .  . 

To find the Contre of Gravity. .  . 

Practical Questions . 

Practical Exercises concerning Ferces .:° , ° » 
On the Motion of Bodies in Fluids . b etait 

On the Motion of Machines, and their Maximum effects 


Pressure of Earth and Fluids against walls and Fortifications— 


Theory of Magazines, &c. . : Pes Wa 
Theory and Practice of Gunnery. Reg igh 2 : 


Promiscuous Problems and Exercises in Mechanics, Statics, Dynam- 


ics, Hydrostatics, Hydraulics, Projectiles, &c. &c. 
Additions : ( ; : : . “ ‘ A 
DescriptiveGeometry .  . SNR Ue ee 
Tables of Logarithms, Sines, and Lingehts . a : 


eo? A 
COURSE 


OF 


MATHEMATICS, éc. 





_ PLANE TRIGONOMETRY CONSIDERED ANALYTICALLY. 


Art. 1. 


"Tuere are two methods which are adopted by mathemati- 
cians in investigating the theory of Trigonometry : the one 
Geometrical, the other Algebraical. In the former, the vari- 
ous relations of the sines, cosines, tangents, ‘&c. of single or 
multiple arcs or angles, and those of the sides and angles of 
triangles, are deduced immediately from the figures to which 
the several enquiries are referred ; each individual case re- 
quiring its own particular method, and resting on evidence 
peculiar to itself: In the latter, the nature and properties of 
the lnear-angular quantities (sines, tangents, &c.) being first 
defined, some. general relation of these quantities, or of them 
in connection with a triangle, is expressed by one or more al- 
gebraical equations ; and then every other theorem or pre- 
cept, of use in this branch of science, is developed by the 
simple reduction and transformation of the primitive equation. 
Thus, the rules forthe three fundamental cases in Plane Tri- 
gonometry, which are deduced by three independent geome- 
trical investigations, in the first volume of this Course of Ma- 
thematics, are obtained Wgebratcally, by forming, between the 

Vor. I. 2 three 


2 ANALYTICAL PLANE TRIGONOMETRY. 


three data nd the three unknown quantities, three equations, 
and obtaining, in expressions of known terms, the value of 
each of the unknown quantities, the others being exterminat- 
ed by the usual processes. Each of these general methods has 
its peculiar advantages. The geometrical method carries con- 
viction at every step; and by keeping the objects of enquiry 
constantly. before the eye of the student, serves admirably to 
guard him against.the admission of error: the algebraical me- 
thod, on the contrary, requiring little aid from first principles, 
but merely at the commencement of its career, Is more pro- 
perly mechanical than mental, and requires frequent checks 
to prevent any deviation from truth. The geometrical me- 
thod is direct, and rapid in producing the requisite conclusions 
at the outset of trigonometrical science; but slow and cir- 
cuitous in arriving at those results which the modern state of 
the science requires: while the algebraical method, though 
sometimes circuitous in the developement of the mere ele- 
mentary theorems, is very rapid and fertile in producing those 
curious and interesting formule, which are wanted in the 
higher branches of pure analysis, and in mixed mathematics, 
especially in Physical Astronomy. ‘This mode of developing 
the theory of ‘Trigonometry is, consequently, well suited for 
the use of the more advanced student ; and is therefore in- 
troduced here with as much brevity as is consistent with its 
nature and utility. 

2. To save the trouble of turning very frequently to the 

Ist volume, a few of.the principal definitions, there given, are 
- here repeated, as follows : 

The sine of an arc is the perpendicular let fall from one 
of its extremities upon the diameter of the circle which passes 
through the other. extremity. — 

The costne of an arc, is the sine of the complement of 
that arc, aod is equal to the part of the radius comprised be- 
tween the centre of the circle and the foot of the sine. 

The tancenT of an arc, is aline which touches the circle 
-in-one extremity of that arc, and is continued from thence till 
it meets a line drawn from or through the centre and through 
the other extremity of the arc, 

. The secant of an arc, is the radius drawn through one of 
‘the extremities of that arc and prolonged till it meets. the tan- 
gent drawn from the other extremity, - 

The versep sine of an arc, is that part of the diameter of 
the circle which lies between the Ceetnntng of the arc and the 
foot of the sine. 

‘The corancent, cosecant, and coversep sine of an are, 
are the tangent, secant, and versed sine, of the complement 
of such arc. — 3. Since 


ANALYTICAL PLANE TRIGONOMETRY, 


3. Since arcs are proper and adequate measures of plane 
angles, (the ratio of any two plane angles being constantly 
equal to the ratio of the two arcs of any circle whose centre 
is the angular point, and which are intercepted by the lines 
whose inclinations form the angle), it is usual, and it is per- 
fectly safe, to apply the above names without circumlocution 
as though they referred to the angles themselves ; thus, when 
we speak of the sine, tangent, or secant, of an angle, we mean 
the sine, tangent, or secant, of the arc which measures that 
angle; the radius of the circle employed being known. 

4. It has been shown in the Ist vol. (pa. 382), that the tan- 
gent is a fourth proportional to the cosine, sine, and radius ; 
the secant, a third proportional to the cosine and radius ; the 
cotangent, a fourth proportional to the sine, cosine, and ra- 
dius ; and the cosecant a third proportional to the sine and 
radius. Hence, making use of the obvious abbreviations, and 
converting the analogies into equations, we have , 





rad. Xsine rad. Xcos. rad. 
tan. = ——_——_,, cot. = ——-—, sec, = —-, cosec, = 
; cos. - sine C086 4.3 
rad2 


———. Or, assuming unity for the rad. of the circle, these 


io sin. Cos. 1 
will become tan. = ——...cot. =——.... sec. = --—. 
cos. sin. COS. 
1 
cosec. == ——. 


These preliminaries being borne in mind, the student may 
pursue his investigations. 2 . 


5. Let asc be any plane triangle, of | (eon ® 
which the side sc opposite the angle a - 
is denoted by the small letter a, the side _ 6 a 
ac opposite the angle 8s by the small 
letter 6, and the side aB opposite the Ast ae 


angle c by the small letter c, and cb perpendicular to as ; 
then is, c=a. cos. B+. cos. A. 

Fer, since ac=6, ap is the cosine of a to that radius ; con- 
sequently, supposing radius to be unity, we have ap=8. cos. 
a. In like manner it is Bp=a.cos.s. Therefore, anv-+sp 
=as=c=a.cos.s-+b.cos. a. By pursuing similar reason- 
ing with respect to the other two sides of the triangle exactly 
analogous results will be obtained. Placed together, they will 
be as below : , 


b=a.cos.c+ ec. cos. A. 


= 6.e0s.c +c. cos. B. 
(f.) 
¢==a.cos.pB +b. cos. A, 


6. Now, 


4 ANALYTICAL PLANE TRIGONOMETRY. 


6. Now, if from these equations it were required to find. 
expressions for the angles of a plane triangle, when the sides 
are given; we have only to multiply the first of these equa- 
tions by a, the second by 6, the third by c, and to subtract” 
each of the equations thus obtained from the sum of the other 
two. For thus we shall have 














3 62 4 a3 
B ss —a?=2be . cos. A, whence cos. 4 = oe : } 
x Seve f | ane : a2 net h2 
@? -+-c2 — }2 =2ac ~COS.B,. - . + COS. B, = oe. (II.) 
; “a : 2 h2 c= 62 
a -+- 62 —c2?=2ab.cos.t,. . . . COS. Cs "ose haath 


. More convenient expressions than these will be deduc- 
‘ed hereafter : but even these will often be found very con- 
venient, when the sides of triangles are expressed in integers, 
and tables of sines and tangents, as well as a table of squares, 
(like that in our first vol.) are at hand. 

Suppose, for example, the sides of the triangle are a=320, 
b = 562, c = 800, being the numbers given in prop. 4, pa. 
161, of the Introduction to the Mathematical Tables : then 
we have : | 
b?4+-c?a? = 853444... . . Nog. = 5-9311751 
Rhee es 889200 8 OO TS Togs = 59538080 
The remainder being log. cos. a, or of 18°20! = 9:9773671 
ene a? -- ¢2—-§2 = 426556. . « log, = 5-6299760 

Nett atop = 512000. . . log, = 5°7092700 — 
The sealer being log. cos. B, or of 33°35’ = 9- 9-9207060 

Then 186° — (18° 20° 433° 35')=128° S = c: c: where all 

the three triangles are determined 1 in 7 lines. 


8. If it were wished to get expressions for the sines, in- 
stead of the cosines, of the angles; it would merely be ne- 
cessary to introduce into the preceding equations (marked II), 
instead of cos. a, cos. B, &c. their equivalents cos. a = 4/ 
(i—sin?. AY, cos, B==,/ (1—sin?. 8), &c. For then, after a 
litle reduction, there would result, 


“sin. = dey RAR Taare Fae (EE Te) 
sin. 2 = = tev BAH FRE ATFO TO) 
Or, paielnine the expression under the radicad into its: tpur 


constituent factors, substituting s for a-rb-re, and Bedueings 
the equations will become 


sin. 


ANALYTICAL PLANE TRIGONOMETY. p 


ain. A=E/is(48 —4) (Js) Gs—e) | 
sin. Bi 4/28(38— 4) (g8—4) (48~—c) (III) 
sin. = 88 —¢) (g8—6) (98 = ¢) 


‘These equations are moderately well suited for computa- 
tion in their latter form ; they are also perfectly symmetrical : 
and as indeed the quantities under the radical are identical, 
and are constituted of known terms, they may be represented 
by the same character ; suppose k: then shall we have 


Pa 


i ak . sin. B ogee . sin. < ae 
sin. A gorse 2 on Ce » (tit.) 
Hence we may SCRE ae deduce a very important theo- 
rem: for, the bbe of these equations, divided by the eon 


st a and the first divided by the third gives aa 
Ee PNG : sin. c 








ve wit 

s— 

gl sin 
a ea 

==—: whence we have 
c 


sin. Ai sin. B.sin.caa@:b:c¢... (IV.) 


Or, in words, the sides of plane triangles are proportional to 
the sines of their opposite angles. (See th. 1, Trig. vol. i). 


9. Before the remainder of the theorems, necessary in the 
solution of plane triangles, are investigated, the fundamental 
‘proposition in the theory of sines, &c. must be deduced, and 
the method explained- by which Tables of these quantities, 
confined within the limits of the quadrant, are mede-to ex- 
tend to the whole circle, or to any number of quadrants what- 
ever.. In order to this, expressions must be first obtained for 
the sines, cosine, &c. of the sums and differences of any two 
arcs or angles. Now, it has been found (1) thata = 6. cos. 
sin. B 


‘sia. A 





c-++c..cos. B. And the equations (1V) give b=a : 


e=a, Substituting these values of b and c for them 





in the tsenie equation, and multiplying the whole by = is = 





it will become 
: sin. A== Sin.-B . COs. ce sin. € . COS. B. 

But, in every plane triangle, the sum of the three angles 

is two right angles; therefore B and c are equal to the sup- 

_ plement of a: and, consequently, since an angle and its sup- 

plement have the same sine (cor. 1, pa. 378, vol. i), we have 

sin. (B+c) = sin. B . cos. c+ sin. ©. cos. B. ‘i 
4 5 


6 ANALYTICAL PLANE TRIGONOMETRY. 


10. If, in the last, equation, c bécome cultvaetieans then 
would sin. c manifestly. become subtractive also, while the 
cosine of c would not change its sign, since it would still con- 
tinue to be estimated on the same radius in the same direction, 
Hence the preceding equation. would become., 
Ue si (acy = sin. B. Cos. c— sin. € . Cos. B, 

11. Let c’ be the complement of c, and 10 he the quarter 
ef the circumference; ‘then will c fee, —c. sin. C= Cos. c, 
and cos. ¢ = sin. oo But (art. 10), sin. (# — c’) = sin. B, 
Cos. a alc c cos. 8. Therefore, substituting for sin. c’, 
cos. c’, their values, there will result sin. (B—c') = sin. B. 
sin. c.— cos. B . cos. c. but because c’ = 1 © —c, we have 


sin. Osage = sin. (s-+co—1Q) ='sid: [(B ee! c) Wig oO} = 


sin. [10 — (s+0)) = —cos.(B-+-c). Substituting this vidoe 
of ie oni in the equation above, it becomes cos. ere) 
= COS. B . COS. C. — SiN. B. Sin. C. 


12. In this.latter equation, if c be made subtractive, sin. c 
will become — sin. c, while cos. c will not change ; conse- 
quently the equation will be transformed to the following, 
Viz. COS. (B ~C) == COS. B. cos. C + Sin. B. sin. ©. 

If, instead of the angles 8 and c, the angles had been a and 
B; or, if a and B represented the arcs which measure those 
angles, the results would evidently be similar: they may 
therefore be expressed generally by the two following equa- 
tions, for the sines and cosines of the sums or differences of 
any two arcs or angles : ei 

sin. (ats) = sin. a. cos. B. ct sin. B. cos, a. V.) 

cos. (A+B) = cos. a. cos. B.. = sin. 4. sin. B. oa ( 

13, We are now in a state to trace completely the muta- 
tions of the sines, cosines, &c. as they relate to arcs in the 
various parts of acircle ; and thence to perceive that tables 
which apparently are included within a quadrant, are, in fact, 
applicable to the whole circle. 

Imagine that the radius mc of the circle, in the marginal 
figure, coinciding at first with ac, turns about the point c (in 
the same manner asa rod would turn ona pivot), and thus 
forming successively with 4c all 
possible angles: the point ™ at 
its extremity passing over all 
the points of the circumference 
aBa’B A, or describing the whole. 
circle. Tracing this motion at-- 
° tentively, it will appear, that at 
the point a, where the arc is 
_ nothing, the sine is nothing also, 
while the cosine does not differ 





é 


ANALYTICAL PLANE TRIGONOMETRY. 4 


from the yates: As the radius we recedes from Ac, the sine 
em keeps increasing, and the cesine cr decreasing, till the 
i op point m has passed over a quadrant, and arrived at 
: in that case pm becomes equal to cp the radius, and thé 
dane cP vanishes. The point m continuing its motion be- 
yond z, the sine pm’ will diminish, while the cosine cP’, which 
now falls on: the contrary side of the centre c will increase. 
In the figure, P m’ and cp’ are respectively the siné and cosine. 
of the arc a’m’ or the sine and cosine of aBM’, which is the 
supplement of a’m’ to 1© half the circumference > whence it 
follows that an obtuse angle (measured by an arc greater than 
a quadrant) has the same sine and cosine'as its supplement; the 
cosine however, being reckoned subtractive or negative, be- 
cause it is situated contrariwise with regard to. the centre c. 
_ When the describing point has passed over 10, or half 
the circumference, and has arrived at a’, the sine p’m vanishes, 
or becomes nothing, as at the point a, and the cosine is again 
equal to the radius of the circle. Here the angle acm has 
attained its maximum limit; but the radius cm: may still be 
supposed to continue its motion, and pass éelow the diameter 
aa’. The sine, which will then be p’m”, will consequently 
fall below the diameter, and will augment as m moves along 
the third quadrant, while on the contrary cp’? the cosine, will 
diminish. -In this quadrant too, both sine and cosine must be 
considered as negative : the former being on a contrary side 
of the diameter, the latter a contrary side of the centre, to 
what each was respectively in the first quadrant. At the point 
B, where the are is three-fourths of the circumference, or 
20, the sine p” mu” becomes equal to the radius cp, and ne 
cosine cP” Lae Finally, in the fourth quadrant, from zB’ 
to a, the sine Pm”, ‘, always below aa’, diminishes in its pro- 
gress, while ihe 2 cosine cP”, which is then found on the same 
side of the centre as it was im the first quadrant, auguments 
till it becomes equal to the radius ca. Hence, the sine in this 
quadrant is to be considered as negative or subtractive, the 
cosine as positive. If the motion of m were continued through 
the circumference again, the circumstances would be exactly 
the same in the fifth quadrant:as in the first. in the sixth as in 
the second, in the seventh as in the third, in the eighth as in 
the fourth ; and the like would be ‘the case in any subsequent 
revolutions. 

Mm. the mutations of the tungent Be traced in like’ man- 
ner, it will be seen that its magnitude passes from nothing to 
infinity in the first quadrant ; becomes negative, and decreases 
from infinity. to nothing in the second; becomes. positive 
again, and increases from nothing wg infinity i in the third quad- 

rant 4 


ANALYTICAL PLANE TRIGONOMETRY. 


rant; and.lastly, becomes BES again, and decreases from 
iafinity to nothing, in the fourth quadrant. - 

15. These conclusions admit of a ready confirmation ; and 
others may be deduced, by means of the analytical expres- 
sions inarts.4 and12. Thus, if a be supposed equal to iQ, 
in equa. Vv, it will DECemR oy 

cos. (LQ -E B) = cos. 1 ©. cos. B = sin. } © - sin. B, 
sin. (10 +8) =sin. 1 i QO. cos. B =£ sin. B «cose 1 Q. 
Bat sin, 10 = rad. = 1; and cos.3 OO: - 

so that nevenors equations will become 

COs.(2 1) cb. B) = sins B 

sin. GO +8) =cos. x. We 
ae which it. is obvious, that if the sine and cosine of an 
arc, less than a quadrant, be regarded as positive, the cosine 
of an arc greater than } © and less than 3 0. will be negative, 


but its sine positive, if B also ve made=10; then shall 
we have cos. O =—1; s.3 O =0.- 
Suppose next, that in the equa. v,A=10; then shall we 
obtain. 7 > 
cos. (4 O45 =—cos. B. 


sin. (4 Oxzs= sin. B; 
which indicates, that every arc comprised. between 4 LO abd 
20, or that terminates in the third quadrant, will ‘have its 
sine and its cosine both negative. In this case too, when 
B=1 Oa or the arc terinmnatig at the ane ef the thine quad- 
rant, we shall have cos. ? © = 0, sin. =— 1. 

Lastly the case remains to be eer in Shitty A=i0 
or in which the arc terminates in the fourth quadrant. < Here 
the Prive equations (V). give | 

cos. (2 O *% 8) = ct sin. Be 
Ne, nee amy (20% 8) =—cos. Bee 
30 Shak in all arcs between ? © and Or the cosines are. posi- 
tive and the sines negative. 

16. The changes of the tangents, with regard to Sositire 
and negative, may be traced by the application of the pre- 
ceding results to the algebraic expression for the tangent ; viz. 


sin. 
tan. et koi For itis hence manifest, that when the sine and 
cos. 


cosine are either both positive or both negative, the ta wngent 
will be positive ; which will be the case in the first and third 
quadrants. But when the sine and cosine have different signs, 
the tangents will be negative, as in the second and fourth : 
quadrants. pa he algebraic expression for the cotangent, viz. 


_ 608. 
cots = re , will produce exactly the same results. 
sl : 


a he 


ANALYTICAL PLANE TRIGONOMETRY. 9 


The expressions for the secants and cosecants, viz. sec. = 
1 1 : 
pa RRRe. show, that the signs of the secants are the 


same as those of the cosines; and those of the cosecants the 
same as those of the sines. 

The magnitude of the tangent at the end of the first and 
third quadrants will be infinite ; because in those places the 
sine is equal to radius, the cosine equal to zero, and therefore 
ae = co (infinity). Of these, however, the former will be 


reckoned positive, the latter negative. 


17. The magnitudes of the cotangents, secants, and cose- 
cants may be traced in like manner ; and the results of the 
13th, 14th, and 15th articles, recapitulated and tabulated as 
below. 

0° 90° 180° 270% 2609 


Sin. O R 0 —R 0 
Tan. O fo) 0 —€ f) 
Sec. R CD as FEN tS eat OD R (VI ) 
Cos. Rk QO —R 0 R 
Cot. © QO —s 0 co) 
Cosec.cO R —O —R co 


The changes of signs are these. 


gin. Cos. tan. cot. sec. COsec. 


ist. 5th. 9th. 13th.) ¥ aT SOY Sage ee 
Qd. 6th. 10th. 14th. ( § +—-—-——-+ (VII.) 
3d. 7th. 11th. 15th(¢g -—-—-++—— 
4th. 8th. 12th. 16th, 3, —-——+-— ; 


We have been thus particular in tracing the mutations, both 
with regard to the value and algebraic signs, of the principal 
trigonometrical quantities, because a knowledge of them is 
absolutely necessary in the application of trigonometry to the 
solution of equations, and to various astronomical and physi- 
cal problems. 

18. We may now proceed to the investigation of other ex- 
pressions relating to the sums, differences, multiples, &c. of 
arcs ; and in order that these expressions may have the more 
generality, give tothe radius any value R instead of confining 
it to unity. This indeed may always be done in an expres= 
sion, however complex, by merely rendering all the terms 
homogeneous ; that is, by multiplying each term by such a 
power of ras shall make xt of the same dimension, as the terms 

an the equation which has the highest dimension. Thus the ex- 
pression for a triple arc. 

Vor. II. 3 sin, 


10 ANALYTICAL PLANE TRIGONOMETRY. 


sin. 3a = 3sin. a—4 sin ?. a (radius = 1) 

_ becomes when radius is assumed = pr, 

~R? sin. 34 = R? 3 sin. a—4 sin?. A 
: - 3p? sin, a—4 sin? . a 
<< -OY sin, SA= . 

R? ; 

_ Hence then, if consistently with this precept, rn be placed 
for a denominator of the second member of each equation v 
(art, 12), and if 4 be supposed equal to B, we shall have 


: sin. A. COS. A sin. A COS. A 
sin. (A-Fa) = te 


te 
R s 


Aes 2 sin. A. COS. A 
That is, sin. 2a=--——-— 
3 | R , . 
And, in like manner, by. supposing B to become successively 
equal to 2a, 34, 4a, &c. there will arise. 
sin. A. cos. 2a-+- cos. A. Sin. 2a. )} 
R 
sin. A. COS. SA-+ cos. A. Sin. 3a. 
ST Ga | ORME CARE EE (VIII.) 
R 
sin. A. COS. 44-- COS. A. Sin. 4A. | 


, R 
_ And, by similar processes, the second of the equations just 
referred to, namely, that for cos. (a-++8), will give successively, 
__cos? . A— Sin? . A 
































sin. 34 = 














sin. 44 = 

















sin. 54 = 
































cos. 2A = 
Aas R 
oo  @0S. A. COS. 2A— sin. A. Sin. Za 
Cis. FA SS abstr <i “Ue 
R 
‘ ee Ex IX. 
OS. A. COS. 3A— sim. A. Sin. 3A f ( X.) 
COs. 4A So a ee a oe A 
R 
C08. A. COS. 44— Sin. A. sin. 4a 
C8. Oe, Soar a PN 


19. If, in the expressions for the successive multiples of 
the sines, the values of the several cosines in terms of the 
sines were substituted for them; and a like process were 
adopted with regard to the multiples of the cosines, other 
expressions would be obtained, in which the multiple sines 
would be expressed in terms of the radius and sine, and the 
_ taultiple cosines in terms of the radius and cosines. 

Assinea=s oe ee , 
sin. 24 = 2s,/R? —s? ; ' ; 
gin. 3a = 3s—4s5 | 
| sin, 4a =(4s—8s?),/n? — 8? b (%) 

. sin. 5a = 5s — 20s? -+4-16s5 | | 

J 


gin. 6a = (6s = 3283-3285), /n2 —s? &. Cos. 





ANALYTICAL PLANE TRIGONOMETRY. 11 


Cosi =e. &, Hii 

cos. 24 = 2c2— 1 

cos. dA =. 403-— 3o, 0 eres 

cos. 4a = 8c4— 8c2 + 1 t (XE); 
cos. 54 = 16c8 —20c? + 5c 


cos. 64 = 3266 — 48c4 + 18c2 —1. 
| Ror Werke i. i ‘ 
Other very convenient expressions for multiple arcs may 
be obtained thus : | | 
Add together the expanded expressions for sin. (B -} 4), 
sin. (B—a), that is, : 


add - - sin. (B--A) = sin. B. cos. A-+ COS. B. Sin. A, 
to  - -. sin. (B—a) = sin. B. cos. A— COS. B. SIN. A; 
there resultssin. (B-+-a) -+ sin. (B—a) = 2 cos.4 . sin. B: 
whence, - , sin. (B--A) = 2 os. a. sin, B—sin. (B — A). 


Thus again, by adding together the expressions for cos. (B-+a) 
and cos. (B—a), we have 
cos. (B-+ A) -+ cos. (B—A) = 2 cos. a. COS. B; 
whence, cos. (B-Fa ) = 2 cos. a . co’. B—COS. (B—A). 
Substituting in these expressions for the sine and cosine of 
B-+A, the successive values a, 24, 3a, &c. instead of B; the 
following series will be produced. | 
sin. 24 = 2 cos. A. Sin. A. 
sin. 3a = 2 cos. a. sin. 24 — sin. a. 
sin. 44 = 2 cos. A. sin. 3a—sin. 2a. (z-) 
sin. mA = 2 cos, A. sin. (n — 1) amsin.(n—2)a. 
cos. 2a = 2 cos. A. cos. A-——cos. 0 (=1). 
COS. SA = 2 cos. A. COS. 24 — COS. A. (xi.) 
cos. 44 = 2 cos. a. cos. 34 — cos. 2a. ee a 
cos. nA = 2 €08. A. cos. (n — 1) a—cos.(n—2)a. 
Several other expressions for the sines and cosines of mul- 
tiple arcs, might readily be found: but the above are the 
most useful and commodious.._~ eee 3 


> > PS 


2 sin A.COSA 





20. From the equation sin. 24 == — , it will be 

easy, when the sine of an arc is known, to find that of its 

half. For, substituting for cos. a its value ,/(r? — sin?), 

_ @sin. a eas sin? A). This 
squared gives Rn? sin? 2a—=4nR? sin? a—4 sin‘ a. 

Here taking sin. a for the unknown quantity, we have a quad- 


there will arise sin, wy Ee 





* Here we have omitted the powers of r that were necessary to render all the 
terms homologous, merely that the expressions might be brought in upon the page 5 
but they may. easily be supplied, when needed, by the rule in art. 18. F 

ratie 


{2 ANALYTICAL PLANE TRIGONOMETRY. 


ratic equation, which solved after the usual manner, gives 
sin. A=  ,/iR? + in \/R? —Sin? 24 
If we make 2a = a’, then will a =.’ and consequently, the 
last equation becomes 
sin. da’ = + ./irn? £ dR yr? —sin? a’ (XII.) 
or sin. 2a’ = + 44/2n? + 2p cos. a’: 
by putting cos. a’ for its value ,/x? — sin? a’ multiplying the 
quantities under the radical by 4, and dividing the whole se- 
cond number by 2. Both these expressions for the sine of 
half an arc or angle will be of use to us as we proceed. 
21. If the values of sin. (a-+s) and sine (a—s), given by 
equa. v, be added together, there will result 
ty : 2 sin. A . COs. 
sin. (a-++-B) + sin. (a—B) = a irl 
sin. a . cos. B=Ir sin. (A+B)-+}R Sin. (a—B) . (XIII.) 
Also, taking sin. (a—s) from sin. (a++B) gives 
2 sin. B. COs. A 








; whence, 





sin. (a-+B) — sin. (a—B) = 








; whence, 





sin. B . cos. A=}R sin (A-+-B)—3R . sin (a—B) . . (XIV.) 
When a=s both equa. x11 and xiv, become 7 
cos. A. Sin. Aa=1R sin. 2a. . (XV.) 
22. In like manner, by adding together the primitive ex- 
pressions for cos. (a+8), cos. (a—B), there will arise 
cos. (a-++B) ++ cos. (A—B) = — ; whence, 
Cos. A. COS. B=1R . cos. (A-+B)-+ IR . cos. (a—B) (XVI.) 
And here, when a=s, recollecting that when the arc is no- 
thing the cosine is equal to radius, we shall have 
cos? a=i1r . cos. 2a+4R?... (XVII. 
23. Deducting cos. (a-+ 8B) from cos. . (a—s), there will 
remain. 


2 sin. a. sin. 
cos. (A—B) — cos. (A-++B) = ee whence, 


sin. A. sin. B=ELR . CoS. (A—B)—1R. Cos. (a+s)(XVIIT.) 
When a=s, this formula becomes 

sin? a=1n2?—41R . cos. 24... (XIX.) 

24. Multiplying together the expressions for sin. (a + 8) 

and sin. (a—B), equa. v, and reducing, there results 

sin. (a-+B) . sin. (A—B) =sin? a—sin? zB. 
And, in like manner, multiplying together the values of cos. 
(a-+-B) and cos. (4—8), there is produced 

Cos. (A-++B) . COS. (A—B) = Cos? a—cos? zB. 

Here, since sin? a—sin? 8, is equal to (sin. a-+sin. B) X (sin. 

7 ea 


ANALYTICAL PLANE TRIGONOMETRY. 13 


4—sin. 8), that is, to the rectangle of the sum and differ- 
ence of the sines ; it follows, that the first of these equations 
converted into an analogy, becomes 

sin. (A— 8B): sin. A — sin. B: ; sin. A+ sin. B: sin. (A-+ 8B) 
(XX.) That is to say, the sine of the difference of any two 
arcs or angles, is to the difference of their sines, as the sum of 
those sines 1s to the sine of their sum. 


If a and B be to each other.as n-+-1 to n, then the preced- 


‘ing proportion will be converted into sin. a : sin. (n-+1) a— 


sin. na: : sin. (n-+1) a+ sinvna: sin. (2n+-1)a.... (XXI.) 

These two proportions are highly useful in computing a 
table of sines ; as will be shown in the practical examples at 
the end of this chapter. i) 

25. Let us suppose a+B=a', and a—s=p ; then the half 
sum and the half difference of these equations will give re- 
spectively a=4(a-+s), and s=}(a—s’). Putting these va- 
lues of a and B, in the expressions of sin. a. cos. B, sin. B . 
COS. A, COS. A. COS. B, Sin. A. Sin. B, obtained in arts. 21, 22, 
23, there would arise the following formule : 


sin, } (a'++B/). cos 3(a'—B’ )= 2 (sin. a’ sin. B’), 
sin. 1 (4' ~B’) . cos 3(A’-+B')= 3 +n (sin. a’— sin. B’), 
cos. a 


(a'-+B’) . cos Han) = 3R cos. a -+ cos. B’), 
& (a’-+B’). sin 1(4'—B')=s 1 Cos. B’— Cos. A). 


Bivitite the second of these formule by the first, there will 
be had 


sin, 4(4'—B’) cos. 3(a’-+B’) __ sin. }(a'—B’) cos. 3(a’-bB’) 











sip. 1 L(a-+e") bel): cos. i 1(4/—s') —B ey ~ cos. 3(4’—B’) : sin. 2(a’+B’) 
__sin. 4 a’ — sin. B sin. tan. ‘COS. Rue, 
——-—. But since —-= ——, and —— = —-, it 
sin. A “sin. B cos. R sin. tan. 


follows that the two factors of the first member of this equa- 
tion, are 
tan. 1(4’—B’) 








respectively ; so that the 


and 

, tan. d(a'+B ve 
s\A —B s / Si ? 

equation manifestly becomes ~~ # 3(4 mec bp In. A —sin. B 


tan. 2(4 (a’-+R’) sin. soles Bf 
. (XXIT.) 


‘This equation is readily converted into a very ase pro- 








~ portion, viz. The sum of the sines of two arcs or angles, is to 


their difference, as the tangent of half the sum of those arcs or 
angles, is to the tangent of half their difference. 


26. Operating with the third and fourth formule of the 


' preceding article, as we have already done with the first and 


second, we shall obtain 
tan. 


14. ANALYTICAL PLANE ‘TRIGONOMETRY, 











tan. tan. $(a’+’) . tan. 3(4’—B’) __ cos. B’— cos. a’ 
a A I f 
Sapa a ee cos. A’+ COs. B 
In like manner, we have by division, 


sin. a’-} sin, B’__ sin. sin. }(a-+B’) sin. A-+sin. B’ 








Sete ee eae oe = tan. 1(a‘ ) os 
cos, a’ cos. B’ cos. cos. 3(a'-+8') a4 one Cagomas B—COS.a" 
‘gin. A’/— Sin. B sin. A’—sin. +B’ 
= cot.d(a’—B’); —=tan.4(a'—B’).. 
Cos, Cos. 4’-+-C0S.B “COS. .B =COS. A’ 


= cot. 3 (a’--2’). | ; 

cos. a’-+ cos. B cot. (a+ B’) 

cos. B— cos. a’ tan. 4 (a'= B 
Making s=0, in one or other of these expressions, there re- 
sults, 























BINA.) ¢ pe a ae } 
I-foosa. 8 cote Lal 

sin. a’ cot i é “ d 4 ( st 

= col. A= le 

l= cos. a’ e tan. 1a’ rai.) 
i--cos. a’ _ cot. ae 
baie ste 2" == cot? 1a! = — 
I—cos. a’ ™ tan. a 


These theorems wri ‘find their application i in some of the 
investigations of spherical trigonometry. 
27. Once more, dividing the expression for sin. (ats) by 
that for cos. (aR), there results 
_sin, (AEB) _ sin. a . cos. BE Sin. B . Cos. AL 


cos.(A+B) cos. A.cOS.B > SiN. A. Sin. B 


then dividing both numerator and denominator of the second 


i tan. 
fraction by cos. a . cos. B, and recollecting that a = = 

















we shall thus obtain 
tan. (A+B) __ R (tan. a + tan, B) 
R ~ R2-F tan. a. tan. B)* 
R? (tan. a & tan. B) 
R? + tan. a. tan. B° 











or, lastly, tan. (A & 8B) = .(XXIIL) 


( R2 
Also, since cot = ae shall have 


t. (ats) = —— R? __ R? + tan. a ‘tan. B 
cot. (aks) = 

wots A tan. (apy tan Atanas es 
which, after a little reduction, becomes 

cot. A. cot. B+ R? 


cot. (ackB) = . (XXIV.) 


cot. B= cot. a 
23. We might now proceed to deduce expressions for the 
tangents, cotangents, secants, &c. of multiple arcs, as well as 
- some of the usual formule. oF verification in the construction 
of tables, such Be. sin. 


. 


ANALYTICAL PLANE TRIGONOMETRY. 15 


sin. (54¢-+a)-Fsin. (54° —a)~sin. (18°--a) “sin. (18% — a) == 
sin.(90°—a) ; sin.a-Fsin. ey (72°-Fa)=sin. (360 
+-a)-+sin.(72°—a). &c. &c. 

But, as these enquiries would extend this chapter to too 
great a length, we shall pass them by ; and merely investigate 
a few properties where mure than two arcs or angles are con- 
cerned, and which may be of use in some subsequent part of 
this volume. 

29. Let a, 8, c, be in any three arcs or angles, and suppose 
radius to be unity ; then 
fi 8 ee A. Sin. c-+ sin. B. sin in (a-+B-+0) 

; sin. (A+B) 

For, by equa. v, sin. (a-+B-++c)=sin. a. cos. (B-++-c)-4-co8. a. 
sin. (8--c), which, (putting cos. B . cos. c—sin. 8 . sin. c tor 
cos. (B-+c)), is = sin. 4 . cos. B . cos. c—Sin. A. sin. B. sin. 
c-+ cos. a. sin. (B-+c); and, multiplying by sin. 8, and add- 
ing sin. A. Sin. c, there results sin. A. Sin. c+ sin. B. Sin. 
(a+s-+c) = sin. a. cos. B. cos. c. sin. B-+ Sin. A. Sin. Cc. 
cos? B-+cos. a. sin. B.. sin. (B-+-c)= sin. a. cos. 8. (sin. B. 
COS. C-+-COS. B . Sin. c)-Lcos. 4. sin. B. sin. (B-++-c)=(sin.a . 
cos. B++cos. a. sin. B) Xsin. (B--c)=sin. (a+) . sin. (8c). 
Consequently, by dividing by sin. (Gre), we obtain the ex- 
pression above given. 

Tn: a similar manner it may be shown, that 

sin. A. Sin. c—sin. B. sin. (A~B-+Cc) 








30. If a, B, c, p, represent four arcs or angles, then writ- 
ing c-++p for c in the preceding investigation, there will result, 
Anusha aye ee eee, 

sin. (A-+B) 

A like process for five arcs or angles will give 
ah extra Orie == sin.A. Sin.(c ++ D-+-£) ++ sin. B. sin. (atefoto+e). 


sin. (As B) 
And for any number, 4, B, c, rae Cl eae 
sin. A. sin. (c--p +..L) -- sin. B. sin. (A+ B 4-0 
sin(B-+-c-+.... L)= Pail CaP TNLISEE (AEB) ioe” ers: hag 
31. Taking again the three a, B, c, we have 
sin. (B—c) = sin. B . Cos. C—8iN. C . COS. B, 
sin. (Cc—a) = Sin. ¢ . COS. ASIN. A . COS. C, 
sin. (A—8) = sin. A. COS. B—SIN. B. COS. A. 
Multiplying the first of these equations by sin. a, the second 
by sin. 8, the third by sin. c; then adding together the equa- 
tions thus transformed, and reducing ; there will result, 
gi. A. Sin. (p—c)-fsin. B.sin.(c—a)-+Ssin, ¢ . sin. (a—b)=0. 
cos. A. sin. (B—c)-}-cos.B. sin. Polen c. sin. (A—B)=0. 
These 








(’ e 


ae 


, 


16 ANALYTICAL PLANE TRIGONOMETRY. 


These two equations obtaining for any three angles what- 
ever, apply evidently to the three angles of any triangle. 
32. Let the series of arcs or angles a, B,c,D....L, 
contemplated, then we have (art. 24), 
sin. (A+B) . Sin. (a—s) = sin? a—sin2 B, 
sin. (B-+c) . sin. (B—c) = sin? B—sin? c, 
sin. (c-++D) . sin. (c~p) = sin? c—sin? p, 
&c. &c. &c. 
sin. (L--a) . sin. (L—a) = sin? L—sin? A. 
If all these equations be added together, the second member 
of the equation will vanish, and of consequence we shall have 
sin. (a+B) . sin. (A—s)-+sin. (B--c) . sin. (B—c)+&c. .. 
» ++» Sin. (L--a)-+sin. (L—a)=0. 
Proceeding in a similar manner with sin. (a—s), cos. (a-++B), 
sin. (B—c), cos. (B-+c), &c. there will at length be obtained 
cos. (A--B) . sin. (A--B)-+cos. (B-++c) . sin. (B—c)-+ &c... 
Deiat tak ak + cos. (L--a) . sin, (L—a)=0. , 
33. If the arcs, a, B, c, &c..... L. form an arithmetical 
progression, of which the first term is 0, the common differ- 
ence pb’, and the last term 1 any number n of circumferences ; 
then will B—a=p’, c—s=p’, &c. at+z=p,; s-+c=3pn, &c.; 
and dividing the whole by sin. pv’, the preceding equations will 
become 
sin. p’-+ sin. 3p’ + sin. 5p’ + &c. = 0, 
cos. D’ -+ cos. 3p -+ cos. 5p’ 4- &c. = 0. Grane, 
If &' were equal 2p’, these equations would become 
sin. p-sin. (p-+ Er )-+sin. (p'+2e')-+sin. (p’+3e')+&c.=0, 
COs. D'--CoSs. (p +E')-+ cos. (p’-+2e )-+ cos. (p’-+-3e’)+&c.=0. 
34. The last equation, however, only shows the sums of 
sines and cosines of arcs or angles in arithmetical progres- 
sion, when the common difference is to the first term in the 
ration of 2.to 1. To investigate a general expression for an 
infinite series of this kind, let 
s-++ sin. a-+sin. (a-+B) + sin. (4-28) sin. (a-++3B) + &c. 
Then, since this series is a recurring series, whose scale of 
relation is 2 cos. B—1, it will arise from the developement of 
a fraction whose denominator is 1—2z . cos. B -+- 22, making 
z=1, 
Now this fraction will PS Sine ili odie aatawisad CO te da void 
1—2z . cos. B-+-22 
Therefore, when z—1, we have 
__ sin. asin. (A-+-B)—2 sin. A. COS. B 
: 2——2 COS. B. 
2 sin. a, COs. B=sin. (a-+B)-fsin. (A=—B) (art. 21), itis equal 
te 


; and this, because 


ANALYTICAL PLANE TRIGONOMETRY, 17 


to sin. 4—sin. (a—B) 

: 2(1—cos. B) . 
2(a’-bB). Sin (a!) by. art. 25, it follows, that sin a — sin 
(a=B) = 2 cos (a — 48) sin 1B; besides which we have 1— 
cos B= 2 sin? 1B. Consequently the preceding expression 


becomes s= ‘sin A-- sin (4-+B).-- sin (a-+2B) + sin no {a-t2B) 
+ &c. ad infinitum == coe ies ee . (XXVE) > 


, ; . F st z é > abe P \ “ 
But, since sin. A — sin, B’ = 2 cos. 





2 sin iB: 
35. To find the sum of nit terms of this series, we have 
_ simply to consider that the sum of the terms past the (n-b 1) th, 
that is, the sum of sin. [a+(-+1)s]+sin. [a-E(n-+2)e]+ 
sin. [4 ++ (n+ 3)8] + &e. ad infinitum, is, by the preceding 
cos.[a-+-(n +4 je] 
~ @sin. 4B 
the former expression, there will remain, sin. a+ sin. (4-+-B) 
+ sin. (a + 28) + sin. (A+ 3p) +... . sin. (a+ mB) = 
cos (a—3B)— cos [a+-(n44)B] _ sin. sin. (af ms) sin. 3(nt lB 


“2 sin. 1B 3B sin. ip 








theorem, = . Deducting this, therefore, from 


(XXVIT.) 
By like means it will be found, that the sums of the cosines 
of arcs or angles in arithmetical progression will be cos. a-- 
cos. (ars) + cos. (A-+ 2B) + cos. (a + 3p) + &c. ad infi- 
sin. GaP). 
it ae sa XXVIUI. 
nium = - aaa: et ) 
Also, 
cos A+ cos Cathy ++ cos (a+ 2p) ++ cos (A fs oP) +, 
. (008 apne) = 08 seen ae AE Seeley ONIX, 
; mm 4B 
36. With regard to the tangents of more than two arcs, 
the following property (the only one we shall here deduce) 
is a very curious one, which has not yet been inserted in works 
of Trigonometry, though it has been long known to mathe- 
maticians. Let the three arcs a, 8, c, together make up 
the whole circumference, OQ: then since tan (a + 8) = 
R?(tan a-b tan B) 
R?— tana. tan B 
+ tan c) =r? X[tan a-+ tan B— tan Gial= ==R? X (tan a-+ 
- «RP eet a+ tan B), 
— tana. tan B’ 
duction, “to tan a. tang. tan c, since tan c=tan[Q— (a+ 
ss 4 is Rr? (tan ae tan B) 
ee am oot Sa ratte rR? — tan a. tan B’ 
ceded in this article. The result therefore i is, that the sum 
VoLs 4 of 





—(by equa. xxi11), we bave R? X (tan abtan B 





tan B— —)==, by actual multiplication and re- 











, by what has pre- 


(8 ANALYTICAL PLANE TRIGONOMETRY. 


of the tangents of any three arcs which together constitute a cir- 
cle, multiplied by the square of the radius, is equal to the yee 
| dust. of those tangents. ... . (XXX.) 


‘Since both arcs in the second and fourth cavoaliints have) 
Laie tangents considered negative, the above property will > 
apply to arcs any way trisecting a semicircle ; and it will 
_ therefore apply to the angles of a plane triangle, which are, 
together, measured. by arcs constituting a semicircle, So 
that if radius be considered as unity, we shall find that, the 
sum of tangents of the three angles of any plane triangle, as 
equal to the continued product of those tangents. (XXXI.) 


37. Having thus given the chief properties of the sines, 
tangents, &c. of arcs, their sines, products, and powers, we 
‘shall merely subjoin investigations of theorems for the 2d and 
3d. cases in the solutions of plane triangles. Thus, with re- 
spect to the second case, where two sides and their included 
angle are given : 

By equa. 1v,a:6::sin a: sin 8. | 

i Ee aa ga ath: amb: : sin a+ sins : sin A— sinB. 

. But, eq. xxir, tan! (a te aye tani (a —8B):: sina 4+ sin 5 B: 
sin asin ; yee ex equal ane a—6b::tani(a "3 B): 
tan }(a—B).... (XXXIE) 

Agreeing mn the result of the iNnataas investigation, 
at pa. 386, vol. 1. 


38. If, instead of having the two aides a, b. given, we know 
their logarithms, as frequently happens in geodesic opera- 
tions, tan (a — B) may be readily determined without first 
finding the number corresponding te the logs of a and &. For 
if a. and b were considered as the sides of a right-angled tri- 
‘angle, in which ¢ denotes the angle, opposite the side a, then 


would tang =>. Now, since a is supposed greater than 6, 
this angle will be greater than half a right angle, or it will be 
measured by an arc greater than } of the circumference, or 
tan ¢— tan 3O sO, 


than 10. Then, because tan (-10)= I-Ftanetan }O 7 10° 





because taniO=R = 1, we have 
tan @-IO)=G ig (io 


@ a—b 


b Ta+b. 


ab tan 1 Ace) b)_tan 3(4—») hi. : consequently 
a+b tani(a+s) cot ic ; 
Ba aan cot 3¢. tan (9— ols a ag 
! ‘rom 


ANALYTICAL PLANE TRIGONOMETRY. 19 


From this equation we have the following practical rule. 
Subtract the less from the greater of the given logs, the re- 
mainder will be the log tan of an angle: from this angle take 
45 degrees, and to the log tan of the remainder add _ the log 
cotan of half the given angle ; the sum will be the log tan of 
half the difference of the other two angles of the plane tri- 
angle. | | 
_ 39, The remaining case is that in which the three sides of 
the triangle are known, and for which indeed we have already 
obtained expressions for the angles in arts..6 and 8. . But, as. 
neither of these is best suited for logarithmic computation, 
(however well fitted they are for instruments of investigation), 
another may be deduced thus: in the equation for cos. a, 
| : . b2-+-c2 -a? , ns 

Asan if we substi- 
tute, instead of cos. a, its value, 1—2 sin? 14, change the 
signs of all the terms, transpose the 1, and divide by 2, we 


a2 — $2 ~c2-+9 Qh bp \2 
shall have sin? ip eee ce ata ey 








' (given equation m1), viz. cos. a= 








Abc aC Abe 
‘Here, the numerator of the second member being the pro- 
duct of the two factors (a--b—c) and (a—b-+-c), the equa- 
) Hatb—c) t@—b+e) pe 
7 yp ° uf, 
Hoee ata re —¢)=1(a+-b+c)—c, and }(a—b+c)=1(a+6-+c) 
—b; if we puts =a-+b-+¢, and extract the square root, 
there will result, | : hints Mn ai 
‘ Ig). (15 — 


(is—a). (s—<). 





tion will become sin? a= 


In Hkece igen. 
manner § 812 28 = JS 


-(XXXIV.) 


sin ic = 4f —— 


(a ane ee es 


a } 
(is—a) . Gs—6) 
ab : 


These expressions, besides their convenience for logarith- 
mic computation, have the further advantage of being perfect- 
ly free from ambiguity, because the half of any angle of a 
plane triangle will always be less than a right angle. 
40. The student will find it advantageous to collect into 
one place all those formule which relate. to the computation 
of sines, tangents, &c.* ; and, in another place, those which 





* What is here given being only a brief sketch of an inexhaustible subject ; 

the reader who wishes to pursue it further is referred to the copious Introduction 

* to our Mathematical Tables, and the comprehensive treatises on ‘Trigonometry, 

by Emerson and many other modern writers on the same subject, where he will 
find his curiosity richly gratified, Ree man | By 

oy are 


20 ANALYTICAL PLANE TRIGONOMETRY. 


are of use in the solutions of plane triangles : the former of 
these: are equations V, VIII, IX, X, XI, ©; @t, XII, XIII, XIV, XV, 
XVI, XVI, XVIII, XIX, KX, XX, Ceti, xxI11, xx1v, xxv; the 
datter are equa. Il, IIT, IV, VII, XXXII, XX¥UI, XXXIV. 

To exemplify the use of some of these acumen the follow- 

ing exercises are subjoined. 

EXERCISES. . 
_ Ex. 1. Find the sines and tangents of 15°, 30°, 45°, 60°, 
and 75° : and show how from thence to find the ines: and tan- 
gents of several of their submultiples. 

First, with regard to the arc of 45°, the sine and cosine are - 
manifestly equal; or they form the perpendicular and base 
of aright-angled triangle whose hypothenuse is equal to the 
~ assumed radius. Thing, if radius be r, the sine and cosine of 

45°, will each be = ,/in?—=n,/1=1n,/2. If x be equal to 
1, as is the case with the tables in use, then wie 
sin 45° = cos 45° = 4,/2="7071068. 


sin cos 
tan 45° == =1=-——= cotangent 45°. 
cos sin 


Secondly, for the sines of 60°.and of 30°: since each angle 
in an equilateral triangle contains 60°, if a perpendicular be 
demitted from any one angle of such a triangle on the oppo- 
site side, considered as a base, that perpendicular will be the 
sine of 60°, and the half base the sine of 30°, the side of the 
triangle being the assumed radius. Thus, if it be R, we shall 


have in for the sine of 30°, and ers —in?=1n,/3, for the 
sine of 60°. When r=1, these become 


sin Po san i «Weare Ran sin 60% == cos 30°= -8660254.,. 
Hence, tan 30° = aS aa = bd Sh See 5773503, 
Ci a he aera 
Ge aes aCe ey 
tan 609 a5 = 2h 5% 1.7320508. 





Consequently, tan 60° =3 tan 30°. | 
Thirdly, for the sines of 15° and 75°, the former arc is the 
half of 30°, and the latter is the compliment of that half arc. 
Hence, substituting 1 for r and } ind 34/3, for cos 4, in the expres-— 
sion sin ,a=+1,/2R?+2R cosa... + (equa. x1), it becomes 
sin 15°=1,/2—,7 3= -2588190.. | 
Hence, sin 75°= cos 15° = Jimi e=78) = WIZ = = 

Ae pe ve Ate ; = +9659258.. 


| sin | 2588190_ 7 
c tly o> = 
jonsequen ve tan 15 che “9650258 ~ 2679492. 
‘9659258 : ; 


7 = ries 
And, tan 5 * 55881907 = +3:7320508. : Mise 


ANALYTICAL PLANE TRIGONOMETRY. 21 


Now, -fialei the sine of 30°, those of 6°, 2, and 1°, may 
easily be found. For, if 5a=30°, we shall have, from equa- 
tion x, sin 5A=5 sin a— 20 sin3 +16 siné a: or,if sin A=z, 
this will become 1625 —20x3-4-5e=°5. This equation solved 
by any of the approximating rules for such equations, wall 
give x="1045285, which is the 'sine-of 6°. | ~ 

Next, to find the sine of 2°, we have again, from equation 
x, Sin 3a = 3. sin A—4 sin? a: that is, if x be put for sin 2°, 
3a — 4e3 == IESOS: bl cubic solved, gives ¢ = 0848995 
= sin 2°. 

Then, if s = sin 1°, we ro ee the second of the equa- 


tions marked x, have 2s ,/ f1—s2 == -0348995 ; whence s is 
found = ‘0174524 == sin 1°. 
Had the expression for the sines of bisected arcs been ap- 
“plied successively from sin 15°, to sin 7°30’, sin 3°45, sin 
1°52)’ sin 561’, &c. a different series of values’ might have 
been obtained : or, if we had proceeded from the quinqui- 
section of 45°, to the trisection of 9°, the bisection of 3°, and 
$0 on, a different series still would ‘have been found. But 
what has been done above, is sufficient to illustrate thas method. 
The next example will exhibit a very simple and compendious 
way of ascending from the sines of smaller to those of larger 
arcs. 
Ex. 2. Given the sine af 1°, to find the sine of 2°, and 
then the sines ‘of 3°, 4°, 5°, 6°, 70, 8°, 9°, and 10°, each by 
“a single proportion. | 
Here, taking first the expression for the sine of a double 
arc, equa. x, we have sin 2° = 2sin 1°71 — sin? 1°==034895. 
Then it follows from the rule in equa, xx, that 
sin 1° ; sin 2°—sin 1° : : gin 2°-++sin 1° : sin 3°=:0523360 





sin 2° : sin 3°—sin 1° : : sin 3°--+sin 19 : sin 4°=-:0697565 
sin 8° : sin 49—sin 1° : : sin 49-bsin 1°; sin 5°=:0871557 
sin 4° ; sin 5°—sin 1° : : sin 5°-++-sin 1°: sin 6°=+1045285 
sin 5° : sin 6°—sin 1° ; : sin 69-+sin 1° : sin 7°=="1218693 
sin 6° : sin 79°—sin 1° ;: sin 79-+sin 1°: sin 8°=='1391731 
sin 7° ; sin 8°—sin 1° : : sin 89-++sin 1° : sin 9°=:°1564375 
sin 8° : sin 9°—sin 1° : : sin 99-+sin 1° : sin10°=°1736482 


To check and verify operations like these, the proportions 


should be changed at certain stages. 


Thus, 


sin 1° : sin 3°—sin 29 : : sin 39 + sin 2° : gin’ 52, 
sin 1° :sin 4°—sin 3° ; : sin 4° + sin 3° : sin.7°, 
sin 4° : sin 7°—sin 3° : : sin 79 + sin 3° : sin 10°. 


The coincidence of the results of these operations with the 
‘analogous results in the BRE CECINE will manifestly establish 
the correctness of both. 

Cor, 


22 ANALYTICAL PLANE TRIGONOMETRY. 


Cor. By the same method, knowing the sines of 5° 10°, 
and 15°, the sines of 20°, 25°, 35°, 55°, 65°, &ec. may be 
found, each by. a single proportion. And the sines of 1°, 9°, 
and. 10°, will lead to those of 19°, 29°, 39°, &c. So that the 
sines may be computed to.any arc : and the tangents and other 
- trigonometrical lines, by means of the expressions in art. 4, &c. 
_ Ex. 3. Find the sum of all the natural sines to every mi- 
nute in the quadrant, radius = 1. 

In this problem. the actual addition - all the terms would 
be a most tiresome labour: but the solution by means of 
equation xxvir, is rendered very easy. Applying that theo- 
rem to the present case, we have sin (a + 1ns) =sin 45°, 
_ sin 3(n-+1) 8=sin 45°0'30", and sin} B=sin 30”. Therefore 


- gin 45° sin 452.0' 30” 
: aera ——_—— = 34382467465 the same sum re- 


quired. 

From another method, the investigation of which ts omitted 
here, it appears that the same sum is equal to 4 (cot 30! +- 1). 

Pr. 4. Explain the method of finding the logarithmic, sines, 
cosines, tangents, secants, “ the Batu Eat BAC; cosines, &e. 
~ being known. 

The natural sines and cosines fies computed to the radius 
unity, are all proper fractions, or quantities less than unity, 
‘so that their logarithms would be negative. To avoid this, 
the tables of logarithmic sines, cosines, &c. are computed to 
a radius of 10006000000, or 101° : in which case the loga- 
rithm of the radius is 10 times the log of 10, that is, it is 10. 

Hence, if s represent any sine to radius 1, then 101° xs= 
sine of the same arc or angle to rad 10*°. “And this, i in logs 
is, log. 101% = 10 log. 10 + log, s = 10 + log. Se 

The log cosines are found by the same process, since the 
‘cosines are the sines of the complements. 
~The logarithmic ies Saas for the ee &c. are de- 
duced thus : 





Tan= =rad— sac Theref. log tan = log rad ‘ log sin — Me 


cos = 10+ log Sit olan Cos. 








oot se sats Therf. log. cot=2 log rad = et Sneaeta de cps tan. 
d2 

Sec a, Therf. log s« sec= -2log rad — —log co8=20- log cos. 
a 


yh d 
Coseo=——. Therf. 1. cosec=2 log redlog sin=20—log sin. 


Verséd 


ANALYTICAL PLANE TRIGONOMETY. —¢3- 


Versed sine = Son et ne pad = : tad re 


Therefore, log vers sin = slog 24 2 log sin 4 arc — 10. 











chord? . (2 sini arc)?» 2 X sin? in? 4 arc 


Ex. b, Given the sum of the. natural tangents of the an- 
gles a and B of a plane triangle = 3-1601988, the sum of the 
tangents of the angles B and c == 31°8765577, and the conti- 
nued product, tana. tans. . tan c == 5: 3047057 : to find the. 
angles a, B, andc. 

“Tt has been Jemonohdied in art. 36, that when vadinn's is 
unity, tbe product of the natural tangents of the three angles 
of a plane triangle is equal to their continued product. sHeote 
the process is this: | La 
_ From tan a + tan 8 + tan ¢ = “he 3047057 
Take tana-+tanB .... =3-°1601988 


Remains tac ....... = 21445069 = tan 65° 
From tan a + tan B+ tanc = 5: 3047057 

~ Take tan B tan ¢ ig es Se 887105577 . | 
Refnains tana. 2... . « = 14981480 = tan 56° 
Consequently, the three angles are 55°, 60°, and 65°. 


a 6. There is a plane triangle, whose sides are three 
consecutive terms in the natural series of integer numbers, 
-and whose largest angle is just double the smallest. Requir- | 
ed the sides and angles of that triangle ? 

If a, B, c, be three angles of a plane triangle, a, b, c, the 
sides respectively opposite to a,B,c; and s=atbte. Then: 
from equa. 111 and xxx1v, we have ; 


| sit is sa). Gs—b). cae 
ie). fret | 


La the three sides of the sl tr jangle be ‘represented 
by x, «+1, and z2+2; the angle a being supposed oppo- 
site to the side x, and c opposite “to the side «+2: then the 
preceding oun will become | 


. and sin tc = 4/-—— 





: ican atte vos +3 ahd ee | 
Sim A y.C+2)% p Va : eas : nat Hie 
sin} Jom OTS ena) ie iy | | 

, 4z(a-+-1) 


Assuming these two expressions eaaat to bach there as ‘they 
ought to ‘be, ad the snesHone there results, after a little re- 


duction, /= DA oas 1) , OF 3x(¢—1)= (+2) an equation 
ig - 2? 








whose 


* 


24 ANALYTICAL PLANE TRIGONOMETRY. 


whose root is 4, or — —1, Hence 4, 5, and 6, are the sides of 
the triangle. . ; i 





oo 
sin. B= 4/7; sin 
The abgIes are, A = 41°-409603 = 41°24’ 34’ 
B= 55°:771191 = 55 46 16 18, 

c= = $20. 819206 = = 82 49 9 8. 


Solution to thie Sean by R. Apnatw. 


Lets age be the triangle, having the angle D 
apc double the angle a, produce a8 to p, mak- : 

ing BD = BC, and join cp; and the triangles B 

cBD, acp are evidently isosceles and equian- | 

cular ; therefore Bp or Be is to cp or AC as C 


ac to ap. Now let ap=z, sc=a2—1, ac=z-+1, then ap= 
2x—1, and the preceding stating becomes x— 1: x++-1:: c+ 
1: 2xz—1, which by multiplying extremes and means gives 
Qo2 Behl cg +2a-+-1, and by subtraction x?= 52, or di- 
viding by x, simply x = 5, hence the sides are 4, 5, 6. 

‘The. same Sonclisron 2 is also readily obtained ‘without the 
use of aigebias 

Ex. 7. Demonstrate that sin 18° == cos 72° is=irn(—1-+ 
4/5), and sin 54° == cos 36° is =ir (1-+-4/5). 3 

Ex. 8. Demonstrate that the sum of the sines of two arcs 
which together make 60°, is equal to the sine of an arc which 
is greater than 60°, by either of the two arcs: Ex. gr. sin 
3’ ++ sin 59° 57’ == sin 60° 3’; and thus that the tapics may 
be continued by addition only. 

Ex. 9. Show the truth of the following proportion : As 
the sine of half the difference of two arcs, which together 
make 60°, or 90°, respectively, is to the difference of their 
gines; so is 1 to ,/2, or ,/3, respectively. 

Ex. 10. Demonstrate that the sum of the square of the 
gine and versed sine of an arc, is equal to the aauare, of dou- 
ble the sine of half the arc. 

Ex. 11. Demonstrate that the sine of | an arc is a mean pro- 
portional between half the radius and the versed sine of dou- 
ble the arc. 

Ex, 12. Show that the secant of an arc is equal to the 
sum of its tangent and the tangent of half its complement. 

Ex. 13. Prove that, in any plane triangle, the base is to 
the. difference of the ther two sides, as the sine of half the — 
sum of the angles at the base, to the sine of haif their diffe- 


rence : also, that the base is to the sum of the other two sides 
pau LAS 


SPHERICAL TRIGONOMETRY. —_25 


as the cosine of half the sum of the angles at the hese to the 
cosine of half their difference. 


Ex. 14. How must three trees, a, B, c, be planted, so that 

the angle at a may be double the angle at B, the angle at 8 dou- 

ble that atc; and so that a line of 400 yards may just go 
round them? 


Ex. 15. In a certain triangle, the sines of the three angles 
are as the numbers 17, 15, and 8, and the perimeter is 160. 
What are the sides and angles ? 


Ex. 16. The logarithms of two sides of a triangle are 
22407293 and 25378191, and the included angle, is 37° 20’. 
It is required to determine the other angles, without first find- 
ing any of the sides? 


Ex. 17. The sides of a triangle are to each other as the 


fractions 1, 3, 1: what are the angles ? 


Ex. 18. Show that the secant of 60°, is double the tan- 
gent of 45°, and that the secant of 45° is a mean proportional 
between the tangent of 45° and the secant of 60°. 


Ez. 19. Demonstrate that 4 times ‘the rectangle of the sines 
of two arcs, is equal to the difference of the squares of the 
chords of the sum and difference of those arcs. 


Ex. 20. Convert the equations marked xxxrv into their 
equivalent logarithmic expressions ; and by means of them 
and equa. rv, find the ee of a triangle whose sides are 5, 
6, and 7. 


SPHERICAL TRIGONOMETRY. 


SECTION I. 


General Properties of Spherical Triangles. 


Arr. 1. Def. 1. Any portion of a spherical surface bounded 
by three arcs of great circles is called a Spherical Triangle. 


Def. 2. Spherical Trigonometry is the art of computing the 
measures of the sides and angles of spherical triangles. 
Vor. Il. 5 Def. 


26 SPHERICAL TRIGONOMETRY. 


Def. 3. A right-angled spherical ivigoule: has one right an- 
gle : the sides about the right angle are called legs ; the side 
opposite to the right angle is called the hypothenuse. 


Def. 4. A quadrantal spherical triangle has one side equal 
to 90° or a quarter of a great circle. 


Def. 6. Two arcs or angles, when compared together, are 
said to be aizke, or of the same affection, when both are less 
than 90°, or both are greater than 90°. But when one 3s 
greater and the other less than 90°, they are said to be unlike, 
or of different affections. 


Art. 2. The small circles of the sphere do not fall under 
consideration in Spherical Trigonometry ; but such only as 
have the same centre with the sphere itself. And hence it is 
that spherical trigonometry is of so much use in Practical 
Astronomy, the apparent heavens assuming the shape of a 
concave sphere, whose centre is the same as the centre of the 
earth. 


3. Every spherical triangle has three sides, and tired: an- 
gles: and if any three of these six parts, be given, the re- 
maining three may be found, by some of the rules which will 
be inv estigated i in this chapter. 


4. In plane trigonometry, the knowledge of the three an- 
gles is not sufficient for ascertaining the sides: for in that case 
the relations only of the three sides can be obtained, and not 
their absolute values: whereas, in spherical trigonometry, 
where the sides are eircular arcs, whose values depend on 
their proportion to the whole circle, that is, on the number 
of degrees they contain, the sides may always be determined 
when the three angles are known. Other remarkable differ- 
ences between plane and spherical triargles are, Ist. That 
in the former, two angles always determine the third ; while 
in the latter they neverdo. 2dly. The surface of a plane 
triangle cannot be determined from a knowledge of the angles 
alone ; while that of a spherical triangle always can. 


5. The sides of aspherical triangle are all arcs of great 
circles, which, by their intersection on the surface of the 
sphere, constitute that triangle. ek 


6. The angle which is contained between the arcs of two | 
great circles, intersecting each other on the surface of the 
sphere, is called a spherical angle ; and its measure is the 
same as the measure of the plane angle wnich is formed by 
two lines issuing from the same point of, and perpendicular 
to, the common section of the planes which determine the 
containing 


SPHERICAL TRIGONOMETRY. 27 


containing sides: that is to say, it is the same as the angle 
made by those planes. Or, it is equal to the plane angle 
formed by the tangents to those arcs at their point of inter- 
section. 

7. Hence it follows, that the gira 
ofa spherical triangle pac, and the three 
planes which determine it, form a kind 
of triangular pyramid, scca of which /p/._- 
the vertex c is at the centre of the ("[-~-[.\y 
sphere, the base azc a portion of the 
spherical surface, and the faces. acc, 
AGB, Bac, sectors of the great circles 
whose intersections determine the sides 
of the triangle. 

Def.’6. A line perpendicular to the afaue of a great circle, 

_ passing through the centre of the sphere, and terminated by 
two points, diametrically opposite, at its surface, is called the 
axis of such a circle; and the extremities of the axis, or the 
points where it meets the surface, are called the poles of that 
circle. Thus, Pep’is the axis, and P,P, are the poles, of the 
great circle onp. | 

If we conceive any sumber of less canola each parallel to 
the said great.circle, this axis will be perpendicular to them 
likewise ; and the points p,.r’, will be their poles also. 

8. Hence, each pole of a creat circle is 90° distant from. 
every point in its circumference ; and all the arcs drawn from 
either pole of a little circle to its circumference, are equal to 
each other. 

9. It likewise follows, that all the arcs of great circles 
drawn through the poles of another great circle, are perpen- 
dicular to it : for since they are great circles by the supposi- 
tion, they all pass through the centre of the sphere, and con- 
sequently through the axis of the said circle.. The *sdme 
thing may be affirmed with regard to small circles. 

10. Hence, in order to find the poles of any circle, itis 
merely necessary to describe, upon the surface of the sphere, 
two great circles perpendicular to the plane of the former ; 
the points where these circles intersect each other will be 
the poles required. 

11. It may be inferred also, from the preceding, that if it 
were proposed to draw, from any point assumed on the sur- 
face of the sphere, an arc ofa circle which may measure the 
shortest distance from that point, to the circumference of 
any given circle; this arc must be so described, that its pro- 
longation may pass through the poles of the given circle. 
And conversely, if an arc pass through the poles of a given 

circle, 





28 SPHERICAL TRIGONOMETRY. 


circle, it will measure the shortest distance from any assumed 
point to the circumference of that circle. 3 
12. Hence again, if upon the sides, ac and sc, (produced 
if necessary) of a spherical triangle sca, we take the arcs, cn, 
em, each equal 90°, and through the radii cn, cm (figure to 
art. 7) draw the plane nem, it is manifest that the point c will 
be the pole of acircle coinciding with the plane nam: so 
that, as the lines em, cn, are both perpendicular to the com- 
mon section cc, of the planes acc, sec, they measure, by their 
inclination the angle of these planes ; or the arc nm measures 
that angle, and consequently the spherical angle sca. 


13. It is also evident that every arc of a little circle, de- 
. scribed from the pole c as centre, and containing the same 
number of degrees as the arc mn, is equally proper for mea- 
suring the angle ca ;_ though it is customary to use only arcs 
of great circles for this purpose. 

14. Lastly, we infer, that if a spherical angle be a right 
angle, the arcs of the great circles which form it will pass 
mutually through the poles of each other: and that, if the 
planes of two great circles contain each the axis of the other, 
or pass through the poles of each other, the angle which they 
include is a right angle. ‘ 

These obvious truths being premised and comprehended, 
the student may pass to the consideration of the following 
theorems. | 


THEOREM I. 


Any Two Sides of a Spherical Triangle are together Greater 
- than the Third. | 3 


This proposition is a necessary consequence of the truth, 
that the shortest distance between any two points, measured 
on the surface of the sphere, is the arc of a great circle pass- 
ing through these points. 


THEOREM IL 


The Sum of the Three Sides of any Spherical Triangle is 
Less than 360 degrees. 


. For, let the sides ac, sc, (fig. to art. 7) containing any 
angle a, be produced till they meet again in p: then will the 
arcs DAC, DBC, be each 180°, because all great circles cut each 
other into two equal parts: consequently pac-+ ppc = 360°. 
But (theorem 1) pa and pe are together greater than the 
) third 


SPHERICAL TRIGONOMETRY. 29 


third side aB of the triangle pap; and therefore, since ca + 
cB + pa + vB = 360°, the sum ca + cB + ap is less than 
3602. Q. E. D. 


THEOREM III. 


The Sum of the Three Angles of any spherical Triangle is 
_always Greater than Two Right Angles, but less than Six. 
1, The first part of this theorem is demonstrated 1 in cor. 2 

of THe. 1v. following. 

2. The angle of inclination of no two of the planes can be 
‘$o great as two right angles ; because, in that case, the two 
planes would become but one continued plane, and the arcs, 
instead of being arcs of distinct circles, would be joint arcs of 
one andthe same circle. Therefore, each of the three sphe- 
rical angles must be less than two right angles ; ‘and conse- 
quently their sum less than six right angles. Q. E. D. 


Cor. 1. Hence it follows, that a spherical triangle may have 
all its angles either right or obtuse ; and therefore the know- 
ledge of any two right angles is not ‘sufficient for the determi- 
nation of the third. 


Cor. 2. If the three angles of a spherical triangle be right 
or obtuse, the three sides are likewise each equal to, or greater 
than 90°: and, if each of the angles be acute, each of the 
sides is also less than 90; and conversely. 


Scholium. From the preceding theorem the student may | 
clearly perceive what is the essential difference between plane 
and spherical triangles, and how absurd it would be to apply 
the rules of plane trigonometry to the solution of cases in 
spherical trigonometry. Yet, though the difference between 
the two kinds of triangles be really so great, still there are 
various properties which are common to both, and which may 
be demonstrated exactly in the same manner. Thus, for. ex- 
ample, it might be demonstrated here, (as well as with regard 
to plane triangles in the elements of Geometry, vol. 1) that 
two spherical triangles are equal to each other, ist. When the 
three sides of the one are respectively equal to the three sides 
of the other. 2dly. When each of them has an equal angle 
contained between equal sides: and, 3dly. When they have 
each two equal angles at the extremities of equal bases. It 
might also be shown, that a spherical triangle is equilateral, 
isosceles, or scalene, according as it hath three equal, two 
equal, or three unequal angles : and again, that the greatest 
side is always opposite to the greatest angle, and the least side 

to 


30 _ SPHERICAL TRIGONOMETRY. 


to the least angle. But the brevity that our plan requires, 
compels us merely to mention these particulars. It may be 
added, however, that a spherical triangle may be at once 
right-angled and equilateral ; which can never be the case with 
a plane triangle. s. | 


THEOREM IV.. 


If from the Angles of a Spherical Triangle, as Poles, there 
be described, on the Surface of the Sphere, Three Arcs of 

- Great Circles, which by their Intersections form another 
Spherical Triangle ; Each Side of this New Triangle will 
be the Supplement to the Measure of the Angle which is at 
its Pole, and the Measure of each of its Angles the Supple- 
ment to that Side of the Primitive Triangle to which it ts 
Opposite. 


From s, a, and c, as poles, let the 
arcs DF, DE, FE, be described: and by 
their intersections form another sphe- 
rical triangle peF; either side, as DE, 
of this triangle, is ‘the supplement of 
the measure of the angle a at its pole ; 
and either angle, as p, “has for its mea- 
sure the supplement of the side av. 





Let the sides az, ac, Bc, of the primitive triangle, be pro- 
duced till they meet those of the triangle per, in the points 
I, L,M, N, G, K: then, since the point a ts the pole of the arc 
DILE, the distance of the points 4 and & (measured on an are 
of a great circle) will be 90°; also, since c is the pole of the 
arc EF, the points c and e will be 90° distant : consequently 
(art. 8) the point & is the pole of the arc ac. In like manner 
it may be shown, that F is the pole of ac, and p that of as. 


This being premised, we shall have pt. = 90°, and 190° 
whence pip + 1e = Db + eExr + in = pe +1 = 180°. There- 
fore DE = 180° —11: that 1s, since 1 is the measure of the 
angle Bac, the arc pE is = the supplement of that measure. 
Thus also may it be demonstrated that er is equal the supple- 
ment to mn, the measure of the angle sca, and that pF is equal 
the supplement to cx, the measure of the angle apc: which 
constitutes the first part of the proposition. ae 

2dly. The respective measures of the angles of the triangle 
DEF are supplemental to the opposite. sides of the triangles 
asc, For, since the arcs au and Be are each 90°, therefore 

ig 


SPHERICAL TRIGONOMETRY. 31 


“a8 AL-PBG = GL -+aB = 180°; whence cu =180°—as; that 
is, the measure of the angle p is equal to the supplement ‘to 
az. So likewise may it be shown that ac, Bc, are equal to the 
supplements to the measures of the respectively opposite an- 
glese and r. Consequently, the measures of the angles of 
the triangle pEF are supplemental to the several opposite sides 
of the triangle axc. Q:. ELD. 

Cor. 1. Hence these two triangles are called supplemental 
or polar triangles. 

Cor. 2. Since the three sides pr, er, DF, are supplements 
to the measures of the three angles A,B, c3 it results that 
DE-- EF +pr+a+B+c=3X 180°=540°, But (th. 2%, 
pe + er + pr< 360°: consequently a+s+c>180°. Thus 
the first part of theorem 3 is very compendiously demon- 
- strated. 

‘Cor. 3. This theorem suggests mutations that are some- 
times of use in computation—Thus, if three angles of a 
spherical triangle are given, to find the sides: the student 
may subtract each of the angles from 180°, and the three re- 
mainders will be the three sides of a new triangle ; the angles 
of this new triangle being found, if their measures be each 
taken from 180°, the three remainders will be the respective 
sides of the primitive triangle, whose angles were given. 

Scholium. . Vhe invention of. the preceding theorem is due 
to Philip Langsberg.. Vide, Simon Steven, liv. 3, de la Cos- 
mographie, prop. 31 and Alb. Girard in loc. It is often how- 
ever treated very loosely by authors on trigonometry: some 
of them speaking of sides as the supplements of angles, and 
scarcely any of them remarking which of the several triangles 
formed by the intersection of the arcs Dr, er, pr, is the one 
in question. Besides the triangle DEF, three others may be 
formed by the intersection of the semi- 
circles, and if the whole circles be con- 
sidered, there will be seven other tri- 
angles formed. But the proposition only 
obtains with regard to the central tri- 
angle (of each hemisphere), which is 
distinguished from the three others in 
this, that the two angles a and F are si- 
tuated on the same ‘aide’ of nc, the two B and E on the same 
side of ac, and the two c and p on the same side of as. 





. THEOREM v. 


‘In Every Spherical Triangle, the following proportion obtains, 
wiz, As Four Right Angles (er 360°) to the surface of a 
Hemisphere ; 


32° SPHERICAL TRIGONOMETRY. | 


Hemisphere ; or, as Two Right Angles (or 180°) to a Great 

Circle of the Sphere; so is the Excess of the three angles 

of. the Triangle above Two Right Angles, to the Area of 

the triangle. : ee 

Let azc be the spherical triangle. Com- 
plete one of its sides as Bc into the circle 
BCEF, which may be supposed to hound 
the upper hemisphere. Prolong also, at 
both ends, the two sides as, ac, until they 
form semicircles estimated from each an- 
gle, that is, until Baz = aBp=caF=acp= 
180°. Then will cpr =180°=sre; and 
consequently the triangle arr, on the anterior hemisphere will 
be equal to the triangle scp on the opposite hemisphere. 
Putting m, m to represent the surface of these triangles, p 
for that of the triangle Bar, q for that of can, and a for that 
of the proposed triangle anc. Then a and m’ together (or 
their equal a and m together) make up the surface of a spheric 
lune comprehended between the two semicircles, acp,. ABD, 
inclined in the angle a: a and p together, make up the lune 
included between the semicircles car, cpr, making the angle 
c: aand q together make up the spheric lune included be- 
tween the semicircles scr, Baz making the angle 8. And the 
surface of each of these lunes, is to that of the hemisphere, 
as the angle made by the comprehending semicircles, to two 
right angles. Therefore, putting 1s for the surface of the 

hemisphere, we have | | 





fans ai Wed 2 : ae 

80° :B::48: a+. 

igoane lage raae . 
Whence, 180°: a+zn-+c::48:3a+-m+p-+q=2a+41s; 


and consequently, by division of proportion, 
as 180° : a+Bp-—+c—180° :: 48: 2a+1s—13 = 2a; 
A-+B-+-c — 180° 
° U Gh O es les gal 
Hi 180° : a+ B-+c—180°:: is: a One 1379 a aaa: 
QE. D.* 


Cor. 1. Hence the excess of the three angles of any spheri- 
cal triangle above two right angles, termed technically the 





————= 


* This determination of the area of a spherical triangle is due to Albert Gi- 
rard (who died about 1633.) But the demonstration now commonly given of the 
rule was first published by Dr. Wallis. It was considered as a mere speculative 
truth, until General Roy, in 1787, employed it very judiciously in the great Trigo- 

“nometrical Survey, to correct the errors of spherical angles. See Phil. Trans. 
vol. 80, and the next chapter of this volume. | 


spherical 


SPHERICAL TRIGONOMETRY. 33 


spherical excess, furnishes a correct measure of the surface 
of that triangle. | 

Cor. 2. If w= 3-14 1593, and d the diameter-of the ‘sphere, 
A +B+o— Hes 180° 


then1 is rd?2. = Cees 





he area of the spherical tri- 


angle. 

Cor. 3. Since the length of the’ radius, in any circle, is 
equal to the length of 57-2957795 degrees, measured onthe 
circumference of that circle ? if the spherical excess be mul- 
tiplied by 57-297795, the product will express the surface of 
the triangle in square degrees. — 

Cor. 4. When a = 0, then A -+3-+ c= 180°: and when 
a=1s, then pres ee Fhe Consequently the sum of the 
three angles of a spherical triangle, is always between 2 and 
6 right angles : which is another confirmation of th. 3. 

Cor. 5. When two of the angles of a spherical triangle are 
right angles, the surface of the triangle varies with its third 
angle. And when a spherical triangle has three right angles 
its surface is one eighth of the surface of the sphere. 

Remark. Some of the uses of the spherical excess, in the 
more extensive geodesic operations, will be shown in the fol- 
lowing chapter. The mode of finding it, and thence the area 
when the three angles of a spherical triangle are given, is ob- 
vious enough ; but it is often requisite to ascertain it by means 
of other data, as when two sides and the included angle are 
given, or when all the three sides are given... In the former 
case, let a and 6 be the two sides, c the included angle, and 
cot 3a . cot 3b-+ cos ¢ c 

sin c 
When the three sides a, b, c, are given, the spherical. excess 
may be found by the following very elegant theorem, disco- 
vered by Simon Lhuillier : 


atb+e | atb—c, a-b+e 
4 « tan 4 a | 4 


E the spherical excess : then is cot Le= 














tanie= Ly (tan 
—a-+b+e 
4 


. tan 


), The investigation of these theorems would oc- 





cupy more space than can bes allotted to them i in the present 
volume. j 


THEOREM VI. 


In every Spherical Polygon, or surface included by any num- 
ber of intersecting great circles, the subjoined proportion 
obtains, viz. As Four Right Angles, at 360°, to the Surface 
of a Hemisphere ; ; or, as Two Right Angles, or 180°, toa 
Great Circle of the Sphere; so is the Excess of the Sum 

Vou. Il. 6 of 


34 SPHERICAL TRIGONOMETRY. 


of the Angles above the Product of 180° and Two Less.than 

the number of Angles of the spherical polygon, to its Area. 

- For, if the polygon be’supposed to be divided into as many 
tnemples as it has sides, by great circles drawn from all the 
angles through any point within it, forming at that point the: 
vertical angles of all the triangles. Then, by th. 5, it will be 
as 360°; 4s :: a++-B-b c— 180°: its area. Therefore, put- 
ting p for the sum of all the angles of the polygon, n for their 
number, and-v for the sum of all the vertical angles of its con- 
‘stituent trtangles, it will be, es composition, 
as 360°: 1s: : P-+v—180° mw: surface of the polygon. -But 
vis manifestly equal to 360° or 180° x 2. Therefore, as 

p—(n—2) 180° 
3609: 4s :; ra 2) 180°; 1s, 


: 360° , the area 











oF the polygon. q. E. D. 
or. 1, If w and d represent the same e quantities as in hear: 

5 cor. 2, then the surface of the polygon will be ors 
fe eae Pana) 180° 

an 720° STs Aa ie eee 
Cor. 2. If n°==57-2957795, then will the surface of the 
polygon in square degrees be =R°. (rP— (n—2) 180°). 
’ Cor. 3.. When the surface of the polygon is 0, then p= 
(n—2) 180° ; and when it is a maximum, that is, when it is 
equal to the surface of the hemisphere, then p= (n — 2) 180° 
-+-360° =n. 180°: Consequently p, the sum of all the angles 
of any spheric polygon, is always Jess than 2n right angles, 
but greater than (2n~4) right angles , denoting the number 
of angles of the polygon. 





» GENERAL SCHOLIUM. 
On the Nature and Measure of Solid Angles. 


A Solid Angle is defined by Euclid, that which is made by 
the meeting of more than two plane angles, which are not in 
the same plane, in one point. 

Others define it the angular space comprised between s se- 
veral planes meeting in one point. 

It may be defined still more generally, the angular space 
included between several plane surfaces or one or more curv- 
ed surfaces, meeting in the point which forms the summit of 

the angle. 

According to this definition, solid angles bear just the same 
fi relation to the surfaces which comprise them, as plane angles 
do to the lines by which they are included: so that as in the 
latter, it is not the magnitude of the lines, but their mutual 
inghhpaten, which determines the angle ; just so, in the former 

Nat 


: 


SPHERICAL TRIGONOMETRY. 35 


itis not the magnitude of the planes, but their mutual inclina- 
tions which determine the angles.» And hence. all those ge- - 
ometers, from the time of Euclid down to the present period, 
who have confined their attention principally to the magnitude 
of the plane angles instead of their relative positions, have 
never been able to develope the properties of this class of 
geometrical quantities ; but have affirmed that no solid angle 
can be said-to be the half or the double of another, and have 
spoken of the bisection and trisection of solid angles, even in 
the simplest cases, as impossible problems. = 6 


But all this supposed difficulty. vanishes, and the doctrine of 
solid angles becomes. simple, satisfactory, and universal-in its 
application, by *assuming spherical surfaces for their measure ; 
just as circular arcs are assumed for the measures of. plane 
anglest. Imagine, that from the summit. of a solid angle 
(formed by. the meeting of three planes) as a centre, any 
sphere be described, and that those planes are produced till 
they cut the surface of the sphere; then will the surface of | 
the spherical triangle, incladed between those planes be a 
proper measure of the solid angle made by the planes at their 
common point of meeting; for no change can be conceived in 
‘the relative position of those planes, that is in the magnitude 
of the solid angle, without a corresponding and proportional 
mutation in the surface of the spherical triangle. If, in like 
- manner, the three or more surfaces which by. their. meeting 
constitute another solid angle, be produced till they cut the 
surface of the same or an equal sphere, whose centre coin- 
cides with the summit of the angle ; the surface of the spheric 
triangle or polygon, included between the planes which deter- 
mine the angle, will be a correct measure of that angle. And 


* Circular arcs are not merely assumed to be the measures of plane angles, . 
they are demonstrated to be so. ‘See Sim. Euclid, Prop. 33, Book VI. It ought 
also to be demonstrated that spherical surfaces are the measures of solid angles. 


_ + Tt may be proper to anticipate here the only objection which can be made 

- to this assumption ; which is founded on the principle, that quantities should al- 
ways be measured by quantities of the same kind. But'this, often and positively 
as it is affirmed, is by'no meansnecessary ; nor in many cases is it possible. To 

~ measure is to compare mathematically : and if by comparing two quaiitities, whose 
ratio we know or can ascertain, with two other quantities whose ratio we wish to 
know, the point in question becomes determined: it signifies not at. all whether 
‘the magnitudes which constitute one ratio, are like or unlike: the magnitudes 
which constitute the other ratio. It is thus that mathematicians, with perfect 
safety and correctness, make use.of space as a measure of -velocity, mass as a 

_ measure of inertia, mass and velocity conjointly as a measure of force, space ‘as 
a measure of time, weight as a measure of density, expansion as a- measure of 
heat, a certain function of planetary velocity as a measure of distance from the 

- central body, arcs of the same circle as measures of plane angles; and it is in 
conformity with this general procedure that we adopt surfaces, of the same sphere, 

as measures of solid angles. , saa 


- the 


36 SPHERICAL TRIGONOMETRY. 


the ratio which subsists between the areas of the spheric tri- 
angles, polygons, or other surfaces thus formed, will be accu- 
rately the ratio which subsists between the solid angles, con- 
stituted by the meeting of the several planes or surfaces, at 
the centre of the sphere. 

' Hence, the comparison of solid Hinalée becomes’ a matter of 
great ease and simplicity ; for, since the areas of spherical 
triangles are measured by the excess of the sums of their an- 
gles each above two right angles (th. 5); and the areas of 
spherical polygons of n sides, by the excess of the sum of 
their angles above 2n—4 right angles (th. 6) ; it follows, that 
the magnitude of a trilateral solid angle, will be measured by 
the excess of the sum of the three angles, made respectively 
by its bounding planes, above 2 right angles ; and the magni- 
tudes of solid angles formed by n bounding: planes, by the ex- 
cess of the sum of the angles of inclination of the several 
planes above 2n—4 right angles. | 

As to solid angles limited by curve surfaces, atts as the an- 
gles at the vertices of cones; they will manifestly be measur- 
ed by the spheric surfaces cut off by the prolongation of their 
bounding surfaces, in the same manner as angles determined 
by planes are measured by the triangles or polygons, they 
mark-out upon the same, or an equal sphere. In all cases, 
the maximum limit of solid angles, will be the plane towards 
which the various planes determining such angles approach, 
as they diverge further from each other about the same sum- 
mit: just.as a right line is the maximum limit of plane angles, 
being formed by. the two bounding lines when they make an 
angle. of 180°. The maximum limit of solid angles is mea- 
sured by the surface of a hemisphere, in like manner as the 
maximum. limit of plane angles is measured by the arc of a 
semicircle. ‘I'he solid right angle (either angle, for example, 
of a cube) is} (=}?) of ‘the maximum solid angle: while the 
plane right angle i is half, the maximum plane angle. 

The analogy between plane and solid angles being thus trac- 

ed, we may proceed to exemplify this theory by a few in- 
stances ; assuming 1000 as the numeral measure of the maxi- 

mum solid angle = 4 times 90° solid == 360° solid. 
~— 1, The solid angles of right prisms are compared with great 
facility. For, of the three angles made by the three planes 
which, by their meeting, constitute every such solid angle, 
two are right angles: and the third is the same as the corres- 
ponding plane angle of the polygonal base ; on which, there- 
fore, the measure of the solid angle depends. Thus; with 
respéct to the right prism with an equilateral triangular base, 
each solid angle is formed by planes which respectively make 
angles 


SPHERICAL TRIGONOMETRY. sz 


angles of 90°, 90%, and 60°. Consequently 90°4-90°-+-60° — 

180° = 60°, is the measure of such angle, compared with 360° 
the maximum angle. . It is therefore, one-sixth of the maxi- 
mum angle. A right prism with a square base, has, in like 
manner, each solid angle measured by 90°-+-90°-+-90° — 180° 
= 90°, whichis} of the maximum angle. And thus may be 
found, that each solid: angle of a right prism, with an eauager al. 


triangular base is’} inax.. angle == #4: °1000. 
"square base» 3S pos = 2 -1000.. 
pentagonal base is ~ . = 5% :1000. 
hexagonal asd. = ;4, °1000. 
heptagonal is = 7 +1000. 
octagonal «1s 3 = 6 *1000. 
nonagonal 1 Le des est = ;5 °1000, 
decagonal is? .. = =; :1000. 
undecagonal is. = #5 °1000. 
duodecagonal is 35. = 2 1000. 
m gonal is = ee 1000 


Hence it may be deduced, that each sald anaes of a Kean 
Tar prism, with triangular base, i is half each solid angle of a 
prism with a regular hexagonal base. Each with regular | 

square base = 2 of each, with regular octagonal base, 





_. pentagonal — bos , decagonal, 
MaRaeonal GB ey Sade ibe dh cass. duodeoasanal, 
i, m—~4 

im gonal = m gonal base. 


m—2 
Hence again we may infer, that the sum of all the solid 
angles of any prism of triangular base, whether that base be 
regular or irregular, is half the sum of the solid angles of a 
- prism of quadrangular base, regular or irregular. And, the 
sum of the solid angles of any prism of 
tetragonal base is = 2 sum of angles in prism of pentag. base. 





pentagonal... = 3 MPR ase Sie oe 4 Basaeoural: 
haxagonal . 2. =$ 6 20. 4 2 3. . heptagonal, 

2 -m—2 

m gonal Maroy esas: Clue gees au (m+1) gonal. 


Let us’ compare the solid angles. of the five regular 
bares In these bodies, if m be the number of sides of each 
face ; nthe number of planes which meet at each solid angle ; 
2 © = half the circumference or 180° ; and a the plane angle 

a | : } cos 5 
made by tivo adjacent faces: then we have sin da =——-—— 


I 
sin 5 O 


ab 


This 


38 SPHERICAL TRIGONOMETRY. 


This theorem gives, for the plane angle formed by every two 
contiguous faces of the tetraédron, 70° 31’ 42”; of the hexaé- 
dron; 90° ; of the octaédron, 109° 28’ 18”; of the dodecaeé- 
dron; 116° 33’ 54”; of the icosaédron, 138° 11’ 23”. Butin 
these se apa the number of faces meeting about each solid 
angle, 3,3,4,3, 5 respectively. Consequently the several so- 
ld angles will be determined by the subjoined proportions : 


Solid Angle. ~ 
BG? 2 Se 70°3142” 180° -*¢ 1000: 87:°73611 Tetraédron. 
360° : 3:90° » —180° :: 1000: 250 Haxaédron. 


360° : 4:109°28'18” —360° :: 1000: 21635185  Octaédron. 
360° : 3°116°33'54”—180° :: 1000: 471°395 Dodecaédron. 
360° : 5:138°11°23”— 540° :; 1000: 419°30169 Icosaédron. 

3. The solid angles at the vertices of cones, will be deter- 
mined by means of the spheric segments cut off at the bases 
of those cones ; that is, if right cones, instead of having plane 
bases, had: bases formed of the segments of equal spheres, 
whose centres were the vertices of the cones, the surfaces of 
those segments would be measures of the solid angles at the 
respective vertices. Now, the surfaces of spheric segments, 
are to the surface of the hemisphere, as their altitudes, to the 
radius of the sphere ; and therefore the solid angles at the 
vertices of right cones will be to the maximum solid angle, 
as the excess of the slant side above the axis of the cone, to 
the slant side of the cone. Thus, if we wish to ascertain the 
solid angles at the vertices of the epee and the right- 
angles cones; the axis of the former is} ,/ 3, of the latter, 

1 ./ 2, the slant side of each being unity. Hence, 

. Angle at Vertex. 

ae ret / 3° : 1000 = 133°97464, equilateral cone, 

Ke 14/2 +: 1000 7292-89322, right-angled cone. 

4. Foti “what has been said, the mode of determining the 
solid angles at the vertices of pyramids will be sufficiently ob- 
vious. “If the pyramids be regular ones, if n be the number 
of faces meeting about the vertical angle in one, and a the 
angle of inclination of each two of its plane faces ; ifm be the 
number of planes meeting about the vertex of the other, and 
a the angle of inclination of each two of its faces: then will 
the vertical angle of the former, be to the vertical angle of the 
- Jatter pyramid, as na--(w—2) 180°, to na — (n—2) 180°. 

Ifa cube be cut by diagonal planes, into 6 equal pyramids 
with square bases, their vertices all meeting at the centre of 
the circumscribing sphere ; then each of the solid angles, 
made by the four planes meeting at each vertex, will be 1 of 
* ‘the maximum solid angle; and each of the solid angles, 
at ene, bases of the pyramids, will be 2, of the maximum solid 

A angle 


SPHERICAL TRIGONOMETRY. 39 


angle. Therefore, each solid angle at the base of such pyra- 
mid, is one-fourth of the solid angle at its vertex: and, if the. 
angle at the vertex be bisected, as described below, ethos of 
the solid angles arising from the bisection, will be double of 
either solid angle at the base. Hence also, and from the first 
subdivision of this scholium, each solid angle of a prism, with 
equilateral triangular base, will be half each vertical angle of 
these pyramids, and double each solid angle at their bases. 

_ The angles made by one plane with another, must be as- 
certained, either by measurement or by computation, accord- 
ing to circumstances. But, the general theory being thus ex- 
plained, and illustrated, the further application of it is left to 
the skill and ingenuity of geometers ; the following simple 
example merely, being added here. 

Ex. Let the solid angle at the vertex of a square pyramid 
‘be bisected. 
- Ist. Let a plane be drawn through the vertex and : any two 

opposite angles of the base, that plane will bisect the solid 
angle at the vertex ; forming two trilateral angles, each equal 
-- to half the original quadrilateral angle. 
2dly. Bisect either diagonal of the base, and draw any | plane 
- to pass through the point of bisection and the vertex of the 
pyramid ; such plane, if it do not coincide with the former, 
will divide the quadrilateral solid angle into two equal quadri- 
lateral solid angles. For this plane, “produced, will bisect the 
great circle diagonal of the spherical parallelogram cut off by 
the base of the “pyramid ; ; and any great circle bisecting such 
diagonal is known to bisect the spherical parallelogram, or 
square; the plane, therefore, bisects the solid angle. 


Cor. Hence an indefinite number of planes may be. draw n, 
each to bisect a given eS solid ae 


SECTION II. _ 
"Resolution of Spherical Triangles. 


Tue different cases of spherical trigonometry, like those in 
plane trigonometry, may be solved either geometrically or al- 
gebraically. Weshall here adopt the analytical. method, as 
well on account of its being more compatible with brevity, 
as because of its correspondence and connection with the sub- 

stance 


40° SPHERICAL TRIGONOMETRY. 


stance. of the preceding chapter.* |The intealle doctrine. may 
be eyaprehicnded 3 in the subsequent problems and theorems. 
al PROBLEM 1. 
To Find Equations, from which may be dedaced the solution 
of all the Cases of Spherical Triangles. 


Let asc be a spherical triangle ; ap the tangent, and cp the 
secant, of the arc 4B; au the tangent, and. cE the secant, of 
the arc ac; let the ca- 
pital letters a, B,.c, de- 
note the angles of the 
triangle, and the small G. 
letters a, 6, c, the oppo- 
site sides BC, AC, AB. 
Then the first equations 
in art. 6 Pl. Trig.. ap- 
plied to the two triangles apr, ape, give, for the former, pr? 
= tan? b + tan? c — tan b. tance . cos a; for the latter pr? > 
= sec? b-+ séc?.c — secb.sece.cosa. Subtracting the 
first of these equations from the second, and observing that 
sec? 6 — tan? 6 = R?= 1, we shall have. after a little reduc- 


sin db. sinc’ cos a 
tion, 1-- ——— cos A — -——— =0. Whence the 
cos6.cosc cog6.cose¢c . 


three following symmetrical equations are obtained : 
cos a= cos 6. cose +sinb. sine coal 3 
(1 












cos b =cosa.cosc+sing. sine .cosB 
cos ¢ = cosa.cos6-+ sine. sin b.cos¢ 


THEOREM VIL. 


i Every Spherical. Triangle, the Sines of the Angles are 
Proportional to the Sines of their Opposite sides. 


If, from the first of the equations marked 1, the value of 
cos a be drawn, and substituted for it in the equation sin? a= 
1-- cos? a, we shall have 

cos? a -+ cos?b . cos? c—2 cosacosb.cose 

sin? a = 1— 

sin? 6 sin? c tin ; 

Reducing the terms of the second side of this equation to a com- 

. mon denominator, multiplying both numerator and denominator 
by sin? a and extracting the sq. root there will result sin a = 
7 (tencost a— cos? be~ cos? c+2-cosa.cosé. COS ¢ 2) 


sin aed 
sin a.sinO.sinc. 


* ‘Por the geometrical method, ihe! reader may consult Simson’s or Playfair’s 
Euclid, or Bishop Horsley’s Elementary Treatises on Practical Mathematics: ¢ 
| Here 


SPHERICAL TRIGONOMETRY. _ at 


Here, if the whole fraction which multiplies sin a, be denoted 
by x (see art. 8 chap. iii), we may write sin A= Xk. sin a. 
And, since the fractional factor, in the above equation, con- 
tains terms in which the sides a, 6, c, are alike affected, we 
haye similar equations nde: sin B, and sin c. That is to say, 
we have ; 

sina =k. sin a . So sin B= Kk. sinb.,.sinc=K. Sin ¢. 


sina sing sinc | 
C ent]: = I. 
onsequ ny) ener eee ee a mah is the 


algebraical expression of the theorem. 


"atts THEOREM VIII. 

In Every Right-Angled Spherical Triangle, the Cosine of the 
Hypothenuse, is equal to the Product of the Cosines of the 
Sides Including the right ss. aps 

_ For, if « be measured by 40, its cosine becomes nothing, 

and the first of the equations I Hecumcs cos a=cos 6. cos c¢. 

| E. D, 
be ene THEOREM IX. oe 

In Every Right- Angled Spherical Triangle, the Cosine of ei- 
ther Oblique Angle, is equal to the Quotient of the Tangent 
of the Adjacent Side divided by the Tangent of the Hy- 
Beate 

in the second of the equations I, the preceding value of 
cos a be substituted for it, and for sin a its value tan a. cos a= 

cos a . cos 6.cosc; then recollecting that 1—cos? c=sin? c, 

there will result, tan a . cos c . cos B= sin | c: whence it fol- 

" Yows that, 


: ‘tan c. 
tan @ . cos = tan €, OF COS B = ten 
he a nb 
Thus also it 1s found that cos c = BB? 
tan a 
- THEOREM be 


In Any Right-Angled Spherical Triangle, the Cozing of ¢ one 
of the Sides about the right angle, is equal to the Quotient 
of the Cosine of the Opposite angle divided by the sine of 
the Adjacent angle. 





From th. 7, we have - 





‘i 5 which, when Aisa right 


Again, from th. 9, we 





ue we } sin b 
angle’ becomes simply sin s = ona 


tan b 
have cos c= ——. Hence by division, 
_ tana 


cos c _ __tan Ase sin a@ cosa 


| sin B sin 1 es tan. tana cos 6 ag ; 
Vor, I. q Now, 








~ 


AB? ‘SPHERICAL TRIGONOMETRY. 
Now, th. ives COS Cc. “Therefore yao =cos b; and 
OS € - gin B . 


COSB 
inlike manner, -—— cs 6. 
sin Cc 


THEOREM XI 

fa Every Right- -Angled Spherical Triangle, the Tangent of 

either of the Oblique Angles, is equal to the Quotient of 
the Tangent of the Opposite Side, divided by the sine of 
the Other Side about the right angle. 








b tan c 
= and cos B= 


sin 
. For, since sin 8 = — 
sin - tan a 


3 


gine sindB tana 
we have — eat at 
cOsSB sina "tan c c 
Whenee, because (th. 8) cosa = cos 6 . cos c, and since sin 
a = cos a. tan a, we have 


sind . sin } sin b 1 
tan Be a ta a et pact eS 
cosa.tane cos6h.cosc.tane cos bcosc. tan c 
tan 6 tan ¢ 





Se Tn like manner, tanc= . Q. E.D 
c gin b> 
~ THEOREM XII. . 

in se Right-Angled Spherical Triangle, the Geniiid of the 
Hypothenuse, is equal to the Quotient of the Cotangent of 
one of the Oblique Angles, divided by the Tangent of the 
Other Angle. 
For, multiplying together the resulting equations of the 

preceding wk ey we have 

tan & tan c as 

sin 6 sin ¢ cos6.cose 

But by th. 8, cos b . cos c = Cos a. 





tan B. tan c= 





cot c 
, Or COS A=——-._ Q. E. D. 
a ~ tan B 





1 
Therefore tan B. tan C= 
a Ba! ie 3 cos 


THEOREM XII. 

In Every Right- Angled Spherical Triangle, the Sine of the 
Difference between the Hypothenuse and Base, is equal: to 
the Continued Product of the Sine of. the Perpendicular, 

Cosine of the Base, and Tangent of Half eae Angle Oppo- 
site tothe Perpendicular ; or equal to the Continued Pro- 
~ duct of the Tangent of the Perpendicular, Cosine of the 
 Hypothenuse, and tangent of Balt the Angle Opposite to 

- the Perpendicular*. 








. * This theorem is due to M. Prony, who published it without demonstration 
in the Connaissance des Temps for the year 1808, and mace use of it.in the con- 
struction ofa chart of the course of the Po. 


Here, 


SPHERICAL TRIGONOMETRY. 43 
Here, retaining the same notation, since we have 


mar for the tangents there be 





sin 5 tan c 
sin a =-———., and cos B= 
sin B - tana 


substituted their values i in sines and cosines, there will arise, 


sin db 
sinc. COS @== ces B. cose: sina = cos B. COS ¢ ee 
sin 


Then substituting for sin a, and sin ¢ . cos a, their values m 
the known formula Sees v chap. iii) viz. 





in sin (ac = sina. eed c¢ —cosa.sin C, 
— COS B 
and recollecting that * "= tant ig,’ 
in B 


it will become, sin (a—c) = sin b.cosc. tan 4s, 
which is the first part of the theorem : Noes if in this resuli 


‘we introduce, instead of cos c, its value — ,it will be 





transformed into sin (a—c) = tan b. cosa. oe 18 ; which is 
the second part of the theorem. Q. E.D 
Cor. This theorem leads rnsanilestly. to an analogous one 
with regard to rectilinear. triangles, which, if h, b, and p de- 
note the hypothenuse, base, and perpendicular, ‘and B, P, the 
angles pe peoy opposite to b, p; may be expressed thus : 
h—-b=p.tanir.....h—p=bd. tan 4s. 
These theorems may be found usefulin redticing ‘inclined lines 
to the plane of the horizon. 


PROBLEM II. 


Given the Three Sides of a Spherical Triangle ; it ig required 
to find Expressions for the Determination of the Angles. 
Retaining the notation of prob. 1, in all its generality, we 

soon deduce from the equations marked 1 in that problem, the 
APuOWINE 5 ; VIZ. 


¢ / 


cos a—cos b.cosc , 
cos A SS 
-~- smd. sinc. 
cos b—cos a@.cosc 




















Cos B == i 
| ‘sina. sinc 
C08 c—cos a.cosb 
cos c = ; 
sin a. sin b 


Aé these equations, however, are not well suited for loga- 
rithmic computation ; they must be so transformed, that their 
second members wil cesole into factors. In Ordee to this, 
substitute in the known equation 1— cos a= 2 sin? 3a, the 
preceding value of cos a, and there will result 

Ae Ot es (b-¢) = cos a 
Fi k's sin b. sinc 


But, 


44 SPHERICAL TRIG ONOMETRY. 


But, because cos p’— cos a’= 2 sin 3 (a +8’). 6in} Stes ) 
(art. 25 ch. i), and consequently, 
glee a+b : os a 


cos (6—c) — cos a= 2 sin —_~-— . sin 
il 





we have, obviously, 

: sin Lab eY, sin 1(a-++e — b) 
sin b. sinc 
Whence, making sssa + 6b + c, there results 

sin (j8—6) . sin (38 —¢) 
sin, 6 . sin ¢ : 
sin (}s—qa) . sin Gs—c). 
‘sina. sinc 
| vy ea g SAE (gs—a@) . sin ({s—)) 
Gh ae ak a sinag.snb | 
The expressions for the tangents of the half angles, might 
have been deduced with equal feility 4 ; and we should have 
obtained, for example, 
sin G—>)- sin (g8—¢) (iit) 
“sinds.sini (s—a) — 
Thus again, the expressions for the cosine and cotangent of 
half one of the angles, are 
sin 38 . 8in 3(S—-a@) 
sin 6. sin ¢ 
sin 38 . sin3(s—a) _ 
sin (is— 6). sin (is — cc)" 
The three latter flowing caaries pow the former, by means 


sin? 1 A= 


sin ta = 4/ 





cue . 


} 1 = 
So also, sin is = 4/ 

















tani ZA =f - 


cos ia = re 





cotla = oy = 


of the values tan == eee cot =- = °. (art. 4 ch. ui.) 


Cor. 1. When two of the Ce as b and c, become equal, 
then the expression for sin 14 beconies 
sin, sin (}) Sb) sin 4 
‘snd | sind- 
Cor. 2,. When all the three sides are equal, or a= b = e, 
sin 3a 


sin A= = 


then sin a = 





sin a : 
Cor..3. In this case, if a = =b=c= 90°; then sin 44 = 


2 
av 14/2 == sin 45°: anda = B==C at 99°: 








iyCor: 4a Mh aes b ec == 60°: then sin 2 la= WE =1y3 
ome gin 35°15/51": and A= B= c= 70°31/42", the same as 
the angle between two contiguous planes of a tetracdon. 

or 6. Ifa= pee ¢ were assumed ==120° : then sin2 2A== 


§in 


SPHERICAL. TRIGONOMETRY. A 


sin 60° -~ LB aarp 
sin 120° 1/3 
that no such triangle can be Bedurtacted (conformably to th. 
2); but that the three sides would, in such case, form three 


continued arcs comleais a great circle of the sphere. 





: and a a Be c= 180° : which shows 


~ PROBLEM Tl. 
Given ‘the Three Angles of a Spherical Triangle, to find 
Expressions for the Sides. 
If from the first and third of the equations, marked 1 
(prob. 1), cos ¢ be exterminated, there will result, | 
cos A. sine -- cosc. sin a . cos b = cos a. sin 6. 


But, it follows from th. 7, that sin pe hie a 














. Substitut- 
Sin A he 
g Be a hs 
ing By sin ¢ this value of it, and for rae oun ae , their equi- 


sin A’ sin @’ 
valents cot A, cot a, we shall have, i 
| e0ta.sin c-+ cos c. cos ae =cota. ath b. 
ee cos a ‘sin b sin B 
Now, cot a. sin b ==, sinb = cosa. =F Cosa. 
sin. & sin Qe 











sin A 
(th. 7). So that the preceding equation at length becomes, 
cos A. sinc = cosa. sin 8 — sina .cosc. cos b. 

In like manner, we have, — 
COs B. sinc = cos b.sin aA — sinB . cosc. cos a. 
Exterminating cos 6 from these, there results 
cos A= cosa. sin B Sin C—COS B . COS C. 

So like- 4 cos B = cosh. sin a sin c—cos a. cos ¢. (1V.) 

wise § cos c = cos c. Sin A sin B— COs A. COS B. 

This system of equations is manifestly analogous to equa- 
tion 1; and if they be reduced in the manner adopted 1 in the 
last problem, they will give 


COs 4 1 (atte) . cos 2 1 O+e—8) > 


s la= 
pan ie - sinB. sinc 
é cost (a-+Bs+c) . cos a-+-o—B) 
BOE Saker Sine aioe noe 
fa tae 1 (a-+s+c) . cos} (atB—c) ° 
wll, ne Co sin a. sin B- 


The expression for the tangent of half a side is 
, cos 1 (a-+B-tc) . cos 1 (B-++- coma) 
1 i 
ene eS cos4 (a+c—B) . cos 4 H (a-+B—c) 
The values of the cosines and cotangents are omitted, £6 
save room ; but are easily deduced by the student. | 
Corel. When two of the angles, as 8 and c, become equal, 
cos ta 





when the value of cos 1 Lo becomes co 


~ 
f 


sin B_ Cor; 


a 


46 ‘SPHERICAL TRIGONOMETRY. 





Cor. 2. When a ==8 =c; thencos } SF aly 
sin A 
ECor. 3. When a =3 =c = 90°, eee et 
< Q- 
Cor. 4.Ilfa=s=c= 60°; then cos ie ee cn 
sin 60 


So thata=b=c=0. Consequently no nibh triangle can 
: be constructed : conformably to th. 3. 
eos 60° Rhee 
sin 120°. 2,73. 
14/3 = cos 54° 449", Hence a = b = c = 109° 28) 18%. 
Sehol. If in the preceding values of sin 1a, sin 44, &c. the 
quantities under the radical were negative in reality, as they 
are In appearance, it would obviously be impossible to deter- 
mine the value of sinia, &c. Butthis value is in fact always 
real. For, in general, sin (a —1@) = — cosa: therefore 
j footers —i0)=—cos} peeeine ; a quantity. which 
is always positive, because as 4 -+ 8 -+ c is necessarily com- 
prised between 1 © and 20, we havei (a+s+c)—160 
greater than nothing, and fess thaniQ. Further, any one side : 
of a spherical triangle being smaller than the sum of the other 
two, we have, by the property of the polar triangle (theorem 
4),iQ —a less than3Q —B-+40O—c; whence} (8-+c—a) 
is less than }Q ; and of course its cosine is positive. 


PROBLEM IV. 


Given Two Sides of a. Spherical Triangle and the Ineluded 
Angle to obtain Expressions for the Other Angles. 
1. In the investigation of the last problem, we had . 
cosa .sin¢e = cosa.sinb—cosec.sina, Page: 
and by a simple permutation of letters, we have 
cosB.sinc = cosb. sina —cos¢.sin 6. cosa: 
adding together these two equations, and reducing, we have 
sin ¢ (cos A -+ Cos B) = ae — cos Cc) sin (a-+-): 
Now we have from theor. 7 
gina sine: sin b sin ¢ 
= , and moons 
sina sin c’ sin B sinc 
Freeing these equations from their denominators, and respec - 
tively adding and subtracting. them, there results 
| sin ¢ (sin a + sin B) = sinc (sin a ++ sin b) 
and sin ¢ (sin a — sin 8) = sin € (sin a — sin b). 
Dividing each of these two equations by the preceding, there 
will be obtained . 
sin 4 -+- sinB sinc sina i sin 


en 


cosa-tcose Iecosc” sin (a-+-b) ” 


* Cor. 5. Wf aspac= 120°: then cos a= ——— 

















sin 


_ SPHERICAL TRIGONOMETRY. 47 


sina— sins. sine sina — sinb 
| Cos A-+-cos B ~J—cosc* sin “sin (a-+b) — 
Comparing these with the equations in arts. 25, 26, 27, ch. iii, 
there will at length result 
Paes cos 1(a—b)- 
tan 1 tC OU ky oe 
RD TA: on alarrb) Reo eVk) 
sin 3 sin 3(a-F6, 
"sin }(a--b) 
Cor. When a= He the first of the above equations becomes 
tan a=tan B= cot $c. sec a. 
And in this case it will be, asrad: sin 1c :: sina or sind : 
“gin te. 

And, as Fad : cos 4 or cos B: ;: tana or tanb : tan 4c. 

2. The preceding values of tan $(4-bs), tan 1(a—p) are 
very well fitted for logarithmic computation : it may, notwith- 
standing, be proper to investigate a theorem which will at once 
lead to one of the angles by means of a subsidiary angle. In 

order to this, we deduce immediately from the second equa- 
tion in the investigation of prob. 3, 


~ cota. sind’ 
cot A= SS Cote’ cosb. 
sin Cc 


Then, choosing the subsidiary angle @ so that 
. < tangd=tana.cosc, . 

that i is, finding the angle g, whose tangent is equal to the pro- 
‘duct’ tan a . cos c, which is equivalent to dividing the original 
triangle into two right-angled triangles, the preceding lea 
will become 3 


tan 1(4 — 8B) = cot te 





€ 


cot A=cot c(cotd.sin b—cosb)=— 





Ga sin b —sing.cos 5), 
And this, since sin (b-—@)=cos 4. ae b—sin @ . cos b becomes 


cote 
cot eee ae sin n (b—9).« 
Which i is a very simple and convenient expression. 
_ PROBLEM V.' - 
Given Two Angles of a Spherical Triangle, and the Side 
_ Comprehended between them ; to find Expressions for the 
_ Other Two Sides. 
1. Here, a similar analysis to that employed in the Breced- 
ing problem, being pursued with respect to the equations rv, 
in n prob. 3, will produce the following formule 


_. sin a-+sin b sin ¢ sin a + sin B a 
cos a + cos 6 ~ I-eosesin (ate) ” 
ss sin a — sin bo sinc _ c sin A—sin B 


cosa--cosb A T+ Cos ¢ Sin sin (A ok Bye? 
Whence, 


48 ~ SPHERICAL TRIGONOMETRY. 


Whence, as in prob. 4, we ‘obtain: 


cos. 3(A—B) y 
tan 1(a+6) = tanic. 2 
n 2( + : RAR Ay, a COs’ 1(a-bB) ( ee 
sin }(a—B)( 
sin 1(4-+B) 


. If it be wished to obtain a side at once, by means of a 











tan 1(a—b) = tante. 


ae angle ; then, find g so that ct ttan "3 ; then will 


1 


ae 
got OS gs FOS (s—¢@). 


~ PROBLEM VI. 

Given Two Sides of a Spherical Triangle, and an Angle Op- 
posite to one of them; to find the Other Opposite Angle. 
‘Suppose the sides given are a, 6, and the given angle ps: 

sina. sin B 


sin } 
fourth properionst to sin b, sin B, and sin a. 
PROBLEM VII. 
Given Two Angles of a Spherical Triangle, and a . Side Obie: 
site to one of them ; to find the Side Opposite to the other. 
Suppose the given angles are a, and B, and 6 the given side: 
sin 6. sin a 


sin B 
portional to sin B, sin 6, and sin a. 


ety Scholium. a 

In problems 2 and 3, if the circumstances of the questions 
leave any doubt, whether the ares or the angles. sought, are 
greater or less than a quadrant, or than a right angle, the 
difficulty will be entirely removed by means of the table of 
mutations of signs of trigonometrical quantities, in different 
quadrants, marked vit in chap. 3. In the 6th and 7th_ prob- 
lems, the question Biepenae will often be susceptible of two - 


then from theor. 7, we have sin a =: ; or, sin Asa 








then th. 7, gives sin a = 3 or, sin a, a fourth pro- 








_. * The formule marked VI, Je vit, converted into atialogies. by making the 
denorninator of the second member the first term, the other two factors the ‘se- 
cond and third terms, and the first inept of the equation, the fourth term: of: the 
proportion, as 

“cos ¢(a4b) : cos 4(a—b) +: cot de: tan 4(a-=-8), 
sin 3(a+b) : sin g(a—b) :: cot ge : tan 7(a—B), &ev &e. 
are called the Analogies of Napier, being invented by that celebrated geometer, 
He likewise invented other rules for spherical trigonometry, known by the name 
of Napier’s Rules for the circular parts; but these, notwithstanding their inge- 
nuity, are not inserted here; because they are too artificial to be applied by a 
young computist, to every case that may occur, without considerable danger of 
misapprehension and error. 
These objections to Napiet’s rules do not appear to me to be well founded. 

ADRAIN- 


solutions : 


we 


aS 


SPHERICAL: TRIGONOMETRY. 49 


solutions : by means of the subjoined table the student may 
always tell when this will or will not be the case. 
1. With the data a, 6, and zB, there can only be one solution 


whens =10 Ss right angle), 
or, oS i a<2© >. sre Goat 
B<70.- aShQ. -6>30—4 
B>PTO--.@l$eOQ--. +64 O—a, 
B>PO.---@>30. Jb Sas 


The drlinsic: 18 susceptible ef ty 


S 


ft) forms ‘and solutions, 


when B<1O. SEO. 5 . b<a, 

. B<} 2 ecay CEOD . 6<1iQ—a, 
B>iO ....a<t ‘SS1O—4, 
B>1O . i a>t0 - . 6>a, 
BZOr>10 . -a=10. 


2. With the data Lp B, ind b; the triangle can exist, but in 
one form, 


‘when b=} oO cone quadrant), 


sve ASLO, ~ BRA: | 
bet O fs At OL LOA, 
bedi) ae Oe era. 
b<igC) beh EO) ee: . Bo ae 


a 
‘It is susceptible of two forms, 


WHEN GEO fa 1 Boe ag 
BRIO. F A<1© ve es B>ILO—A,; 
b<10. ~A>i0....B<1O—a, 
i) eet Ls elon: ee rreay Ul 
_b<or>} Gye . As=l 


. ©. 

It may here be observed, that all the analogies and formu- 
le, of spherical trigonometry, i in which cosines or cotangents 
are not concerned, may be applied to plane trigonometry ; » 
taking care to use only a szde instead of the sine or the tan- 
gent of aside; or the sum or difference of the sides instead of 
_ the sine or tangent of such sum or difference. The reason of 
this is obvious: for analogies or theorems raised not only 
from the consideration of a triangular figure, but the curva- 
ture of the sides, also, are of consequence more general ; 
and therefore, though the curvature should be deemed eva- 
nescent, by reason of a diminution of the surface, yet what 
depends on the triangle alone will remain notwithstanding. 

We have now deduced all the rules that are essential in 
the operations of spherical trigonometry ; and explained un- 
der what limitations ambiguities may exist. That the student, 
however, may want nothing further to direct his practice in 
this branch of science, we shall add three tables, in which the 
several formule, already given, are respectively applied to 
the solution of all the cases of right and oblique-angled sphe- 
rical triangles, than can possibly occur. - 


Vor. IE. 8 TABLE 


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SPHERICAL TRIGONOMETRY. 















































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SPHERICAL TRIGONOMETRY. 


54 | 



























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55 


SPHERICAL TRIGONOMETRY, 





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56 SPHERICAL TRIGONOMETRY. 


Questions for Exercise in Spherical Trigonometry. 


Hey, 1. In the right-angled spherical: triangle sac, right- 


angled at a, the hypothenuse a = 78°20’, and one Jeg ¢ = 
76°52’, are given: to find the angles B, and c, and the other 
leg b. 











| ) i sin ¢ 
Here, by table 1 case 1, sin c =——- 3 
ie eS hae 
tan c cosa. 
cos B = 5S oe ee COSH= 


‘tana’? © cos ¢ 
Or, log si sinc =logsinc—logsina-+ 10, 
log cos 8 = log tan c—log tan a -++ 10. 
log cos b = logcos a—log cos ¢ +- 10. 
Hened, 10 -++ log sin c = 10 -+ log sin 76°52’ = 19-9884894 
| log sina = log sin 78°20’ = 9-9909338 


Remains, log sinc = log sin 83°56! = 99985556 











Here c is acute, because the given leg is less than 90°, 


Again, 10 + log tance = 10+ ‘log tan 76°52’ = 20: 6320468 





log tana = log tan 78°20" = 10°6851149 
Remains, logcosp= log cos 27°45'= 99469319 


BIS here acute, because a and ¢ are of like affection. 
coe 10 +-log cos a = 10 -++ log cos 78920' = 19-3058189 
log cos ¢ = log cos 76°52’ = 9°3564426 





= 


Remains, log cos 6 = log cos 27°98’ = 9-9493763 





where 6 is less than 90°, because a and c both are so. 
Ex. 2. Inaright-angled spherical triangle, denoted as above, 
are given a = 78°20’, B = 27°45’; to find the other sides and 


angle. Ans. 6 = 2708’, e = 76°52, c = 83°56", 
Ex. 3. In a spherical triangle, with Aa right angle, given 
= ieeeat c == 31°51’; to find the other parts. 
Ans. @ = 113°55', c= 28°51’, B = 104° 8’. 


. Ex. 42> Given 6. = 27°6', ¢ = 76°52 ; to ftid, the other. 


parts. Ans. a@ = 78°20! 3 = 27°45, c = 83°56", 
’ Ex. 5. Given 6 = 42°12'3 = 48° ; to find the other parts. 
Ans. @= 64°40. or its supplement, 
= 54°44’, orits supplement, 

c= 64°35, or its supplement. 

Ba. iB. Given B == 48°, c = 64935’; required the other 
parts ? Ans. 6 = 42°?) ¢ = 54a e = 64°40, 


SPHERICAL TRIGONOMETRY. BY 


Ex, 7. In the quadrantal triangle anc, given the quad- 
rantal side a == 90°, an adjacent angle c= 42° 1% and the 
opposite angle a == 64° 40’; required the other Hels of the 
triangle - / 


Ex. 8. In an oblique angled spherical triangle are given 
the three sides viz. a = 56° 40’, b = 83° 13’,c = 114° 30’: 
to find the angles. iid 

Here, by the fifth case of the table 2, we have 
sin (38—b) . sin _sin (}s—c). 
sin & . sin ¢ 
Or, log. sin 1 a= log sin (4s—6)-+Flog sin (3s—c) + ar. comp. 

log. sin b + ar. comp. log. sinc: where s=a-+-b-+c. 


sind a= 


log sin (4s—b) = log sin 43° 58! 1 = 98415749 
log sin (1s—c) = log sin 12° 41° 1 = 9°3418385 

A. C. log sin b =a . c. log sin 83°.13' = 0°0030508 .. 
4, c. log sine =a.c. log sin114° 30° = 0-0409771 
Sum of the four logs ..... 19-2274413 


Half sum = log sin 14 = log sin 24° 15’ 1 = 9-6137206 06 
, Consequently the angle a is 482 31’ 


Then, by the common analogy, . | 
As, sina:..sin 56°40’... log = 9:9219401 
To, sina...sin 48°31’... log = 9°8745679 
So is, sith 6b... sin 83°13... . log = 9:9969492 
To, sine. ..sin 62°56)... log = 9:9495770 
And so is sinc... sin 114°30 . . log = 979590229 
To, sinc... sin 125°19'. . . log = 9°9116507 
So that the remaining angles are, B= 62°56, and c == 125°19' 


~ 


@dly. By way of comparison of methods, let us find the 
angle a, by the analogies of Napier, according to case 5 table 
3, In order to which, suppose a perpendicular demitted from 
the angle c on the cpporte side c. ‘Then shall we have tan 4 
tan } (b—a) . tan} ser) 

tan ic 


diff. seg. of c= 


This in logarithms, is. 
_ log tan 1 (b--a) = log tan 69°56’ £ = 10-4375601 
log tan 3 1 (b—a) = log tan 13°16’ 1 =. 9°3727819 


Their sum = 19-8103420 
Subtract log tan te = log tan 57°15’ = 10°1916394 


Rem. log cos dif. seg = log cos 22°34’ = 9-6187026 
Hence, the segments of the base are 79° 49’ and 34° 41%, 


Veu. I. 9 " . - Therefore 


bo) woh 


58s SPHERICAL, TRIGONOMETRY. 


2 Mbbrelore, since cos a = tan 79° 49’ Xcot b: 
To log tan. ‘adja. seg. = log tan 79° 49’ = 10°7456257 
Add log tan side b = log-tan 83° 13 = 9:0753563 


‘The sum rejecting 10 fromthe index2 
log cos a=log cos 48° 32'§ Te tiee eee 

The other two angles may be found as before. The "he prefer- 
ence is, in this case, manifestly due to the former method. 

Ex. 9. In an oblique-angled spherical triangle, are given 
two sides equal to 114° 40’ and 56° 30’ respectively, and the 
angle opposite the former equal to 125° 20’ to find the other 
’ parts. Ans. Angles 48° 30’, and 62° 55’; side, 83° 12’. 

Ex. 10. Given, in a spherical triangle, two angles, equal 
to 48° 30’, and 125° 207, and the side opposite the latter, to 
find the other parts. 

Ans. Side opposite first angle, 56° 40’; other side, 83° 12’; 
third angle. 62° 54’. 

y Nemes 8 We Given two sides, equal 114° 30’, and 56° 40’; 
and their included angle 62° 54’: to find the rest. 

Ex. 12. Given two angles, 125° 20’ and 48° 30’, and the 
side comprehended between them 83°12": to ine the other 
parts. “ 

Ex. 13. Ina spherical triangle, the atts are 48° 31’, 
62° 56’, and 125° 20’: required the sides ? 

Ex. 14. Given two angles, 50° 12’, and 58° 8' ; ; and aside 
opposite the former, 62° 42’; to find the other parts. 

Ans. The third angle i is either 130°56’ or 156°14’, 
Side betw. giv. angles, either 119°4" or 152214’. 
' .Side opp. 58°87, either 79°12" or 100°48'. 

Ex. 15. The excess of the three angles of a triangle, 
measured on the earth’s surface, above two right angles, is 1 
second ; what is its area, taking the earth’s diameter at7 9572 
miles ? 





- Ans. 76-'75299, or nearly 762 square miles. 
Ex. 16. Petereion the solid angles of a regular pyramid, 
with hexagonal base, the altitude of the pyramid being to each 
side of the base as 2 to 1. - 
Ans. Plane angle between each two lateral faces 126°52'112, 
between the base and each face 66°35'12"1. 
Solid angle at the vertex 114°49768 ‘The max. angje 
Each ditto at the base 22234298 being 1000. 


ON GEODESIC 


“Lae Pp | ’ 


ON GEODESIC OPERATIONS, AND- THE FIGURE. OF 
; THE EARTH. 


SECTION I... 


General Account.of this kind of Surveying. 
Art. 1. In the treatise on Land Surveying in the first 
volume of this Course of Mathematics, the directions were 
‘restricted to the necessary operations for surveying fields, 
farms, lordships, or at most, counties; these being the only 
operations in which the generality of persons, who practise 
this kind of measurement, are likely to be engaged : but there 
are especial occasions when it is requisite to apply the princi- 
ples of plane and spherical geometry, and the practices of sur- 
veying, to much more extensive portions of the earth’s sur- 
face ; and when of course much care and judgment are called 
into exercise, both with regard to the direction of the practical 
operations, and the management of the computations. The 
extensive processes which we are now about to consider, and 
which are characterised by the terms Geodesic Operations and 
Trigonometrical Surveying, are usually eae for the ac- 
complishment of one of these three objects. The finding 
the difference of longitude, between two ab dine distant 
and noted meridians ; as the meridians of the observatories at 
Greenwich and Oxford, or of those at Greenwith and Paris. 
2. The accurate determination of the geographical positions 
of the principal places, whether on the coast or inland, in an 
island or kingdom; with a wiew to give greater accuracy to 
maps, and to accommodate the navigator with the actual posi- 
tion, as to latitude and longitude, of the principal promonto- 
ries, havens, and ports. These haves till lately, been deside- 
rata, even in this country: the position of some important 
points, as the Lizard, not being known within seven minutes 
of a degree ; and, until the publication of the board of Ord- 
nance maps, the best country maps being so erroneous, as in 
some cases to exhibit blundérs of three miles in distances of 
less than twenty. 
3. The 


- 60 TRIGONOMETRIGAL SURVEYING. 


3. T e§ measurement of a degree in various situations ; and 
thence the determination of the figure and magnitude of ‘the 
earth. 

When objects so important as these are to be attained, it is 
manifest that, in order to ensure the desirable degree of cor- 
rectness in the results, the instruments employed, the opera- 
tions performed, and the computations required, must each 
have the greatest possible degree of accuracy. Of these, the 
first depend on the artist ; the second on the surveyor or en- 
gineer, who conducts them ; and the latter on the theorist and 
calculator : they are these last which will chiefly engage our 
oes in the present chapter. 

. {n the determination of distances of many miles, whether 
for the survey of a kingdom, or for the measurement of a de- 
gree, the whole line intervening between two extreme points 
is not absolutely measured; for this, on account of the inequa- 
lities of the earth’s surface, would be always very difficult, 
and often impossible. But, a line of a few miles in length is 
very carefully measured on some plane, heath, or marsh, which 
is so nearly level as to facilitate the measurement of an actual- 
ly horizontal line ; and this line being assumed as the base of 
the operations, a variety of hills and elevated spots are select- 
ed at which signals can be placed, suitably distant and visible 
one from another : the straight lines joining these points con- 
stitute a double series of triangles, of which the assumed base 
forms the first side; the angles of these, that is, the angles 
made at each station or signal staff, by two other signal staffs, 
are carefully measured by a theodolite, which is carried suc- 
cessively from one station to another. In such a series of tri- 
angles, care being always taken that one side is. common to 
two of them, all the angles are known from the observations at 
the several stations, and aside of one of them being given, 
namely that’of the base measured, the side of all the 1 rest, ag 
well as the distance from the first angle of the first triangle to 
any part of the last triangle, may be found by the rules of trigo- 
nometry. And so again, the hearing of any one of the sides, 
with respect to the meridian, being deternfined by observation, 
the bearings of any of the rest, with respect to the same me- 
ridian, will be known by computation.. In these operations, in 
is always adviseable, when circumstances will admit of it, to 
measure another base (called a base’of verification) at or near 
the ulterior extremity of the series : for the length of this base, 
computed as one of the sides of the chain of triangles, com- 

ared with its length determined by actual admeasurement, 
will be a.test of the accuracy of all the operations made inthe 

series between the two bases. 
- 3. Now 


vw. 
oe 


GEODESIC OPERATIONS. » 6 


3. Now, in every series of triangles, where 
each angle is to be ascertained with the same in- 
strument, they should, as nearly as circumstances 
will permit, be equilateral... For, if it were pos- 
sible to choose the stations in such manner, that 
each angle should be exactly 60 degrees; then, .— 
the half number of triangles in the series, multi-: 
plied into the length of one side of either trian- 

.gle would, as in the annexed figure, give at once 
the total distance ; and then also, not only the 
sides of the scale«or ladder, constituted by this 
series of triangles; would be perfectly parallel, 
but the diagonal steps, marking the progress from- 
one extremity to the other, would be alternately 
parallel throughout the whole length. Here too, — 
the first side might be found by a base crossing it - 
perpendicularly of about halfits length, as at a; and the last 
side verified by another such base,r at the opposite extremity. 
If the'respective sides of the series of triangles were 12 or 
18 miles, these bases might advantageously be between 6 and 
7, or between 9 and 10 miles respectively ; according to cir- 
cumstances. It may also be remarked, (and the reason of it 
will be seen in the next section) that whenever only two an- 
gles of a triangle*can be actually observed, each of them 
should be as nearly as possible 45°, or the sum of them about 
90° ; for the less the third or computed angle differs from 90°, 
the less probability there will be: of any considerable error. 
See prob. 1 sect. 2, of this chapter. 

4. The student may obtain a general notion of the method 
employed in measuring an arc of the meridian, from the fol- 
Jowing brief sketch and introductory illustrations. 

The earth, it is well known, is nearly spherical. It may be 
either an ellipsoid of reyolution, that is, abody formed by 
the rotation of an ellipse, the ratio of whose axes is nearly 
that of equality, on one of those axes ; or.it may, approach 
nearly to the form of such an ellipsoid or spheroid, while its 
deviations from that form, though small relatively, may still be 
sufficiently great in themselves to prevent its being called a 
spheroid with much more propriety than it is called a sphere. 
One of the methods made use of to determine this point, is 
by means of. extensive Geodesic operations. 

The earth however, be its exact form what it may, is a 
planet which not only revolves in an orbit, but turns upon 

-an axis. Now, if we conceive a plane to pass through the 
axis of rotation of the earth, and through the zenith of any 
place on its surface, this plane, if prolonged to the limits of 
‘ the 





62 TRIGONOMETRICAL SURVEYING. 


“the apparent celestial sphere, would there trace the circum- 


s 


ference of a great circle, which would be the meridian of that 
place. All the points of the earth’s surface, which have their 
zenith in that circumference, will be under the same celestial 
meridian, and will form the corresponding terrestrial meridian. 
If the earth be an irregular spheroid, this meridian will be a 
curve or double curvature ; but if the earth be a solid of re- 
volution, the terrestrial meridian will be a plane curve. 

5. If the earth were a sphere, then every point upon a ter-, 
restrial meridian would be at an equal distance from’the cen- 
tre, and of consequence every degree upon that meridian 


‘would be of equal length. But if the earth be an ellipsoid 


of revolution slightly flattened at its poles, and protuberant 
at the equator ; then, as will be*shown soon, the degrees of 
the terrestrial meridian, in receding from the equator towards 
the poles, will be increased in the duplicate ratio of the right 


sine of the latitude; and the ratio of the earth’s axes as well 


as their actual magnitude, may be ascertained by comparing 
the lengths of a degree on the meridian in different latitudes. 
Hence appears the great importance of measuring a degree. 
6. Now, instead of actually tracing a meridian on the sur- 
face of the earth,—a measure which is prevented by the in- 
terposition of mountains, woods, rivers, and seas,—a construc- 
fion is employed which furnishes the samewesult. It consists 
in this. pg oe : 
Let ancppr,*&c. be a series of triangles, carried on .as 
nearly as may be, in the direction of the meridian, according 





Rt Lele r 

ai BE hy 
to the observations in art. 3. These triangles are really spheri- 
cal or spheroidal triangles ; but as their curvature is extreme- 
ly small, they are treated the same as rectilinear triangles, ei- 
ther by reducing them to the chords of the respective terres- 
trial arcs ac, aB, Bc, &c. or by deducting a third of the excess, 
of the sum of the three angles of each triangle above two 
right angles, from each angle of that triangle, and working 
with the remainders, and the three sides, as the dimensions 


_ of a plane triangle; the proper reductions to the centre of 


the station, to the horizon, and to the level of the sea, having 
been previously made. ‘These computations being made 
throughout 


7 


GEODESIC. OPERATIONS. gs 


throughout the series, the sides of the successive triangles are 
contemplated as arcs of the terrestial spheroid. Suppose 
that we know, by observation, and the computations which 
will be explained in this chapter, the azimuth, or the inclina- 
tion of the side ac to the first portion am of the measured 
meridian, and that we find by trigonometry, the point m where 
that curve will cut the side sc. The points a, B, cv, being in 
the same horizontal plane, the line am will also be in that 
plane : but, because of the curvature of the earth, the pro- 
longation mm’, of that line, will be found above the plane of the 
second horizontal triangle scp : if, therefore, without chang- 
ing the angle comm, the line mm’ be brought down to coincide 
with the plane of this second triangle, by being turned about 
Bc as an axis, the point m will describe an arc of a circle, 
which will be so very small, that it may be regarded as a 
' right line perpendicular to the ‘plane scp: whence it follows, 
that the operation is reduced to bending down the side mm’ in 
the plane of the meridian, and calculating the distance am’, 
to find the position of the point m’. By bending down thus in 
imagination, one after another, the parts of the meridian on 
the corresponding horizontal triangles, we may obtain, by the 
aid of the computation, the direction and the length of such 
meridian, from one extremity of the series of triangles, to the 
other. 


A line traced in the manner we have now been describing, 
or deduced from trigonometrical measures, by the means we 
have indicated, is called a geodetic or geodesic line ; it has the 
property of being the shortest which can be drawn between 
its two extremities on the surface of the earth ; and it is there- 
fore the proper itinerary measure of the distance between 
those two points: Speaking rigorously, this curve differs a 
little from the terrestrial meridian, when the earth is not a 
solid of revolution: yet, in the real state of things, the dif- 
ference between the two curves is so extremely minute, that 
it may safely be disregarded. 


_7. If now we conceive a circle perpendicular to the eas. 
tial: meridian, and passing through the vertical of the place 
of the observer, it will represent the prime vertical of that 
place. The series of all the points of the earth’s surface 
which have their zenith in the circumference of this circle, 
will form the perpendicular to the meridian, which may be 
traced in like manner as the meridian itself. 


In the sphere the perpendiculars to the meridian are great 
circles which all intersect mutually, on the equator, in two 
points diametrically opposite: but in the ellipsoid of revolu- 

; tion, 


64 TRIGONOMETRICAL SURVEYING. 


tion, and a fortiori in the irregular spheroid, these concurring 
perpendiculars are curves of double curvature. Whatever 
be the nature of the terrestrial spheroid, the parallels to the 
equator are curves of which all the points are at the same 
‘datitude : on an ellipsoid of revolution) these curves are plane 
and circular. 

8. The situation of a wate’ is determined, when we know 
either the individual perpendicular to the meridian, or the in- 
dividual parallel to the equator, on which it is found, and its 
position on such perpendicular, or on such parallel. ‘There- 
fore, when all the triangles, which constitute such a series as 
we have spoken of, have been computed, according to the 
principles just sketched, the respective positions of their an- 
gular points, either by means of their longitudes and latitudes 
or of their distances from the first meridian, and from the per- 
. pendicular to it. The following is the method of computing 
these distances. 

Suppose that the triangles ABC, BCD, ‘&e. (eee the fig. to art. 
6) make part of a chain of triangles, of which the sides are ° 
arcs of great circles of a sphere, whose radius is the distance 
from the level or surface of the seato the centre of the earth ; 
and that we know by observation the angle cax, which measures 
the azimuth of the side ac, or its inclination to the meridian ax. 
Then, having found the excess E, of the three angles of the 
triangle ace (ce being perpendicular to the meridian) above 
two right angles, by reason of a theorem which will be demon- 
strated in prob. 8 of this chapter, subtract a third of this ex- 
cess from each angle of the triangle, and thus, by means of the 
following proportions find ac, and cc. 

sin (90°—1eE : cos aE BAG: AL a 

sin (90°—4e : sin (cac—tE) ¢: AC : CC. 
The azimuth of az is known immediately, because Bax = CAB 
—cax ; and if the spherical excess proper to the triangle ABM’ 
be computed, we shall have 

, amB = 180’—mM4B—aBM +E. 

To determine the sides am’, pm’, a third of & must be deducted 
from each of the angles of the triangle asm ; and then these 
proportions will obtain: viz. | 
sin (180° —m ap — apm’ + 3x’) : sin (anu — 2 LE if . AB AM, 
sin (180° — map — asm’ 2n’) ; sin ‘(map — te’) $3 ap: BM’. 

In each of the right-angle triangles Abs, m ‘dD, are known 
two angles and the hypothenuse, which is all that is necessary 
to determine the sides 4b, 68, and md, dp. Therefore the dis- 
tances of the points zB, p, from the meridian and from the per- 


pendicular, are known. | 
9. Pro- 


GEODESIC OPERATIONS. 65 


9. Proceeding in the same manner with the triangle acn, 
or mpn, to obtain an and pn, the prolongation of cp; and. 
then with the triangle pyr to find the side nr and the angles 
DNF, DEN, it-will be easy to calculate the rectangular co-ordi- 
nates of the point x, 

‘The distance fr and the angles prn, nFf, being thus known, 
we shall have (th. 6 cor. 3 Geom.) | 


fre = 1809 —Erp—prn—nrf. — 


So that, in the right-angled triangle frr, two angles and one 
side are known ; and therefore the appropriate spherical ex- 
Pag hese! be computed, and thence the angle ref and the sides 
je, Fr. Resolving next the right-angled triangle eEP, we shall 
in like manner obtain the position of the point 5 with respect 
to the meridian Ax, and to its perpendicular ay ; that is to say, 
the distances re, and ae=ap—ep. And thus may the compu- 
tist proceed through the whole of the series. It is requisite 
however, previous to these calculations, te draw, by any suit- 
able scale, the chain of triangles observed, in order to see 
_whether any of the subsidiary ‘triangles AON, NFP, &c. formed 
to facilitate the computation of the distances from the meri- 
dian, and from the perpendicular to it, are too obtuse or too 
acute. 

Such, in few words, is the method to be followed, hen we 
have principally i in view the finding the length of the portion 
* the meridian comprised between any two points, as « and 

It is obvious that, in the course of the computations, the 
pith of a great number of the sides of triangles in the 
series is determined ; it will be easy therefore to check and 
verify the work in its process, by comparing the azimuths 
found by observation, with those resulting from the calcula- 
tions.. The amplitude of the whole arc of the meridian mea- 
sured, is found by ascertaining the latitude at each of its ex- 
tremities ; that is, commonly by finding the differences of the 
zenith distances of some known fixed star, at both those ex- 
tremities. 

10. Some ruatitaniaticiiee, employed in this kind of opera- 
tions, have adopted different means from the above. They 
draw through the summits of all the triangles, parallels to the 
meridian and to its perpendicular; by these means, the sides 
of the triangles become the hypothenuses of right- angled tri- 
angles, which they compute in order, proceeding from some 
known azimuth, and without regarding the spherical excess, 
considering all the triangles of the chain as described on a 
plane surface. - This method, however, is manifestly defec- 
tive in point of accuracy. a 

Vou. J. 10 Others 


86 TRIGONOMETRICAL SURVEYING. 


een have computed the sides and angles of all the tri- 
angles, by the rules of spherical trigonometry: Others again, 
reduce the observed angles to angles of the chords of the re- 
spective arches, and calculate by plane trigonometry, from 
such reduced angles and their chords. Either of these two 
methods is equally correct as that by means of the spherical 
excess: so that the principal reason for preferring one of 
these to the other, must be derived from its relative facility. 
As to the methods in which the several triangles are contem- 
plated as spheroidal, they are abtruse and difficult, and may, 
happily, be safely disregarded : for M. Lengendre has demon- 
strated in Mémoires de la Classe des Sciences Physiques et Ma- 
thematiques de l’ Institut, 1806, pa. 130, that the difference be- 
_ tween spherical and spheroidal angles, is less than one siatreth 
of a second, in the greatest of the triangles which occurred 
in the late measurement of an arc of a meridian, between the 
eae of Dunkirk and Barcelona. 

11. Trigonometrical surveys for the purpose of measuring 
a degree of a meridian in different latitudes, and thence in- 
ferring the figure of the earth, have been undertaken: by. dif- 
ferent philosophers, under the patronage of different govern- 
ments. As by M. Mapertuis, Clairaut, &c. in Lapland, 1736 : 
by M. Bouguer and Condamine, at the equator, 1736—1743 ; 
by Cassini, in lat. 45°, 1739—40,; by Boscovich and Le- 
maire, lat. 43°, 1752; by Beccaria, lat. 44° 44’, 1768; by 
Mason and Dixon in " America, 1764—8 ; by Major Lamb- 
ton, in the East Indies; 1803; by Mechain, Delambre, &c. 
France, &c. 1790—1805 ; by Swanberg, Ofverbom, &c. in 
Lapland, 1802; and by General Roy, Colonel Williams, Mr. 
Dalby, and Colonel Mudge, i in England, from 1784 to the pre- 
sent time. The three last mentioned of these gatyeya\are 
doubtless the most accurate and important. 

The trigonometrical ' survey in England was first commenc- 
ed in conjunction with similar operations in France, in order 
to:determine the difference of longitude between the meri- 
dians of the Greenwich and Paris observatories; for this 
purpose, three of the French Academicians, M. M. Cassini, 
Mechain, and Legendre, met General Roy and Dr. (now Sir 
Charles) Blagden, at Dover, to adjust their plans of. opera- 
tion.. In the course of the survey, however, the English phi- 
losophers, selected from the Royal Artillery officers, expand-__ 
ed their views, and pursued their operations, under the pa-. 
tronage, and at the expense of the Honourable Board of Ord- 
mance, in order to perfect the geography of England, and to 
determine the lengths of as many degrees on the meridian as 
fell within the compass of their labours. 

12. It 


"GEODESIC OPERATIONS. 67 


12. It is not our province to enter into the history of these 
surveys: but it may be interesting and instructive to speak a 
little of the instruments employed, and of the extreme accu- 
racy of some of the results obtained by them. | | 

These instruments are, besides the signals, those for mea- 
suring distances, and those for measuring angles. ‘The French 
philosophers used for the former purpose, in their measure- 
ment to determine the length of the zetre, rulers of platina 

_and of copper, forming metallic thermometers. The Swedish 
- mathematicians, » Swanberg and Ofverbom, employed iron 
bars, covered towards each extremity with plates of silver. 
General Roy commenced his measurement of the base at 
Hounslow: Heath with deal rods, each of 20 feet in length. 
Though they, however, were made of the best seasoned tim- 
ber, were perfectly straight, and were secured from bending 
in the most effectual manner ; yet the changes in their lengths, 
occasioned by the variable ‘moisture and dryness of the air, 
were so great, as to take away all confidence in the results 
deduced from them. Afterwards, in consequence of having 
found by experiments, that a solid bar of glass is more dilata- 
ble than a tube of the same matter, glass tubes were substi- 
tuted for the deal rods. They were each 20 feet long, inclosed 
in wooden frames, so as to allow only of expansion or con- 
traction in length, from heat or cold according to a law. 
ascertained by experiments. . The base measured with these 
was found to be 27404-08 feet, or about 5-19 miles. Several 
years afterwards the same base was re-measured by Colonel 
Mudge, with a steel-chain of 100 feet long, constructed by 
Ramsden, and jointed somewhat like a watch-chain. This 
chain was always stretched to the same tension, supported on 
troughs laid horizontally, and allowances were made for 
changes in its length by reason of variations of temperature, at 
the rate of -0075 of an inch for each degree of heat from 62° 
of Fahrenheit: the result of the measurement by this chain 
- was found notto differ more than 22 inches, from General Roy’s 
determination, by means of the glass tubes : a minute differ- 
ence ina distance of more than 5 miles ; which, considering 
that the measurements were effected by different persons, and 
with different instruments, is a remarkable confirmation of the 
accuracy of both operations. And further, as steel chains can 
be used with more facility and convenience than glass rods, 
this re-measurement determines the question of the compara- 
tive fitness of these two kinds of instruments. 
_ 13: For the determination of angles, the French and Swe- 
dish philosophers employed repeating circles of Borda’s con- 
struction : instruments which are extremely portable, and with 
which 


& 


68 TRIGONOMETRICAL SURVEYING. 


which, though they are not above 14 inches in diameter, the 
observers can. take angles to within 1” or. 2‘ of the truth. 

But this kind of instrument, however great its ingenuity in 
theory, has the accuracy of its observations necessarily limited 
by the imperfections of the small telescope which must be 
attached to it. General Roy and Colonol Mudge made use of 
a very excellent theodolite constructed by Ramsden, which, 

having both an. altitude and an azimuth circle, combines the 
powers of a theodolite, a quadrant, and a transit instrument, 

and is capable of measuring horizontal angles to fractions of a 
second. This instrument, besides, has a telescope of a much 
higher rhagaifyine power than had ever before been applied 
to observations purely terrestrial ; and this is one of the supe- 


-riorities in its construction, to which i is to be ascribed the ex- 


treme accuracy in the results of this trigonometrical survey. 
» Another circumstance which has augmented the accuracy 
of the English measures, arises from the mode of fixing and 


using this theodolite.. In the method pursued by the Con- 


tinental mathematicians, a reduction is necessary to the plane 
of the. horizon, and another to bring the observed angles to 
the true angles at the centres of the signals: these reductions, 
of course, require formule of computation, the actual em- 
ployment of which may lead to error. But, in the trigono- 
metrical survey of England, great care has always been taken 
to place the centre of the theodolite exactly in the vertical line, 
previously or subsequently occupied by the centre of the 
signal: the theodolite is also placed in a perfectly horizontal 
position. Indeed, as has been observed by a competent judge, - 
“In no other survey has the work in the field been conducted 

so much with a view to save that in the closet, and at the same 

time to avoid all those causes of error, however minute, that 

are not essentially involved in the nature of the problem. The 

French mathematicians trust to the correction of those errors ; 

the English endeavour to cut them off entirely ; and it can 

hardly be doubted that the latter, though perhaps the slower | 
and more expensive, is by far the safest proceeding.” - 

14. In proof of the great correctness of the English survey, 
we shall state a very few. partichlars, besides es is already 
mentioned in art. 12. _ 

- General Roy, who first measured the base on Liokaisoun 


‘Heath, measured another on the flat ground of Romney- 


Marsh in Kent, near the southern extremity of the first series 


_ of triangles, and at the distance of more than 60-miles from 


the first base. The length of this base ‘of verification, as 
actually measured, compared with that resulting from the 
computation through the whole series of triangles, differed 
only by 28 inches. Colonel 


‘GEODESIC OPERATIONS. 69 


Colonel. Mudge measured another. base of verification on 
Salisbury plain. Its length was 36574-4 feet, or more than. 
7 miles; the measurement did not differ more than one inch 
from the computation carried through the series of triangles 
from Hounslow Heath to Salisbury Plain. A most remarkable 
proof of the accuracy with which all the angles, as well as 
the two bases, were measured ! ; 

. The distance between Beachy-Head. in Sussex,-and Dun- 
nose in the Isle of Wight, as deduced from a mean of four 
series of triangles, is 339397 feet or more than 641 miles. 
The extremes of the four determinations do not differ more 
than 7 feet, which is lessthan 12 inchesin a mile. Instances 
of this kind frequently occur in the English survey.* But 
we have not room to specify more. We must now proceed 
to discuss the most. important problems connected with this 
subject ; and refer those who are desirous to consider it more 
minutely, to Colonel Mudge’s, ‘‘ Account of the Trigonome- 
trical Survey ;”? Mechain and Delambre, ‘* Base du Systeme 
Métrique Décimal ;” Swanberg, ‘* Exposition des Opérations 
faites en Lapponie ;’’ and Puissant’s works entitled ‘ Geode- 
sie’ and ‘* Traite de Topographie, d’Arpentage, &c.”’ 


rd 
SECTION II. 


Problems connected with the detail of Operations in Extensive 
_ .. Trigonometrical Surveys. a Dens 
PROBLEM I. 
It is required to determine the Most Advantageous Conditions 
| : of Triangles. | 
J. In any rectilinear triangle asc, it is from the propor- 

tionality of sides to the sines of their opposite ‘angles, ax : 
Bc : : sin € : sina, and consequently as. sin Cc 
a=se.sinc. Let az be the base, which 

is supposed to be measured without percep- 

tible error, and which therefore is assumed 
as constant; then finding the extremely A’ Pp 
‘Small variation or fluxion of the equation on this hypothesis, 
itis AB.coSA.as=sine.Bc-+Bc.cose.c. Here; since 
_ ® Puissant, in his * Geodésie,” after quoting some of them, says, ** Neanmoins, 
Jusqu’s présent, rienntegale en exactitude les opérations géédesiques qui ont servi 
de fondement & notre systéme.métrique.”’.. He, however, gives no instances. We 
have no wish to depreciate the labours of the French measurers; but we cannot — 
yield them the preference on mere assertion. 





we 


70 TRIGONOMETRICAL SURVEYING. 


we are ignorant of the magnitude of the errors or variations 
expressed by a andc, suppose them to be equal (a probable 
_ supposition, as they are both taken nhl the same Daa 
ipaing ‘eat denoted: by v: then. will 

AB COS A—BC COS C 


Bo=v X ——; 
sinc 7 








Bess dt eRe Fe ae Ce 
or, substituting ris for its equal agi the equation will be- 


come Bo=0 X(Bc . vs —BC. eae) : 
sina Sin © 
or, finally pc» Bc (cot a~ cot c). 

This equation (in the use of which it must be recollected 
that v taken in seconds should be divided by Rr”, that is, by 
the length of the radius expressed i in seconds) gives the error 
Bc in the estimation of Be occasioned by the ‘errors in 
angles a and.c. Hence, that these errors, supposing them to 
be equal, may have no influence on the determination of BC, 
we must have a=c, for in that case the second member of the 
equation will vanish. 

2. But, as the two errors, denoted by A, and ¢, which we 
have supposed to be of the same kind, or in the same direc- 
tion, may be committed in different directions, when the equa- 
tion will be sc = + v . Bc (cot a-++ cotc); we must enquire 
what magnitude the angles a and ¢ ought to have, so that the 
sum of their cotangents shall have the least value possible ; 
for in this state it is avhnifest that Bc will have its least value. 
But, by the formule in chap. 3, we have 
cok fio sin (a+ ¢) C)_ _ sin (a-Fe) ea 

; gin A.sinc cos cos (anc). ye cos (a++c) 

2 sin B 

cos os (Ag nn Sree +-cosB 

2 sin B 
cos (Amc) + cosB 
And hence, whatever be the magnitude of the angle zg, the 
error in the value of sc will be the least when cos 5 wc) is 
the greatest possible, which is when a=c. 

We may therefore infer, for a general rule. that the most 
advantageous state of a triangle, when we would determine one 
side only, is when the base is equal to the side sought. 

3. Since, by this rule, the base should be equal to the side 
_ sought, itis evident that when we would determine two sides, 

the fon geveaagenus condition of @ triangle 4 as shat tt be equi- 
latera . 








Consequently, po = v. BC. 


4. It 


GEODESIC OPERATIONS. vse 


4, It ate happens, however, that a base can be commo- 
diously measured which is as long as the sides sought. Sup- 
posing, therefore, that the length of the base is limited, but 
that its direction at least may be chosen at pleasur: e, we pro- 
ceed te enquire what that direction should be, in the case 
where one only of the other two sides of the triangles is to 
be determined. 

Let it be imagined, as before, that apis the base of the tri- 
angle apc, and se the side required. It is proposed to find 
the least value of cot a = cot c, when we cannot have a = c. 

Now, in the case where the negative sign obtains, we have 
AB~BC.COSB BC-—AB.COSB  AB2—BC* 





— 














cot a—cot c= meen 


| c. sin B AB.SINB AB. BC.SINB: 
This equation again manifestly indicates the equality of az and 
BC, in circumstances where it is possible : but if an and Bc are 
constant, itis evident, from the form of the denominator of the 
last fraction, that the fraction itself will be the least, or cot 

A--cot c the least, when sin B 1s a Ran that is, when 
B=90°. 

5. When the positive sign phi we have 9 afoot = 

/ (Bc* — aB* sin 4) | 

cot a-- Pra samara mae cot a-+ 4/ C xPe 1.) 
Here, the least value of the: expression cnbes: the radical sign, 
is obviously when a = 90°. And in that case the first term, 
cot a, would disappear. Therefore the least value of cot a+- 
cot c. obtains when a = 90°; conformably to the rule given 
by M. Bouguer (Fig. de la Terre, pa. 88). But we have 
already seen that in the case of cot a—cot c, we must have 
B=90°. Whence we conclude, since the conditions a==90°, 
B=90°, cannot obtain simultaneously, that a medium result 
would give a=B. 

If we apply to the side ac the same reasoning as to Bc, simi- 
lar results will be obtained: therefore in general, iilén the 
base cannot be equal to one or to both the sides required, the most 
advantageous condition of the triangle. is, that the base be the 
longest possible, and that the two angles at the base be equal. 
These equal angles however, should never, if possible, be 
less than 23 degrees. 








PROBLEM IL. 


To deduce, from Angles measured Out of one of the stations 
but Near it, the True Angles at the station. 


When the centre of the instrument cannot be placed in the 
vertical line occupied by the axis of the signal, the angles ob- 
served must undergo a reduction, according to circumstances. 
. Let 


72 -'TRIGONOMETRICAL SURVEYING. 


‘4. Let c be the centre of the station, B 
R the place of the centre of the instru- ~ ‘ 
ment, or the summit of the observed an- 
gle ave: it is required to find c, the 
‘Measure of acz, supposing there to be 
known APB == P, BPC==p, cp==d, BC=L, cP 

ACER. 

Since the exterior angle of a triangle i is equal to the sum 
‘of the two interior opposite angles (th. 16 Geom.), we have, 
with respect to the triangle tap, arg = P-}-14P; and with re- 
gard to the triangle sic, ates=¢-+ Cpe. Making these two 
values of ai equal, and transposing rar, there results 

Cm P + IAP —.CBP. 
But the triangles car, cap, give 


cP | d. sin Cas 70) 


sin CAP = = sin IAP = — sin APCS 
AC ie 





d. ae 





CP 
gin CBP = — . sin BPC = — 
BC 


And, as the angles CAP, CBP, ae by the hypothesis of the 
problem, always very small, their sines may be substituted 
for their arcs or measures : therefore | 
_d sin Reed d. sin sop 
R ee ie 
Or, to have the eadutiiow 3 mn gocondes 
d sin (p-Ep) . sit sin p 

sinl” ( R ke 

Phe use of this formula cannot in bay case be embarrassing, 
provided the signs of sin’ p, and sin (P +p) be attended to. 
T'bus, the first term of the correction will be positive, if the 
angle ‘(e--p) is comprised between 0 and 180° ; and it will 
become negative, if that angle surpass 180°. The contrary 
will obtain in the same circumstances with regard to the se- 
cond term, which answers to the angle of direction p. The 
letter rn denotes the distance of the object a to the right, 1 
the distance of the object B situated to the left, and p the angle 
at the place of observation, between the centre of the station 
and the object to the left. | Pe 


P= 








2. An approximate reduction to the centre may indeed be 
obtained by a single term: but itis not quite so correct as the 
form above. For, by reducing the two fractions in the second 
member of the. last equation but one to a common denomina- 
hy the correction becomes 


f du. sin (e-+p)— da. sin up 
eats somragrr) imma 


But 


GEODESIC: OPERATIONS, 73 


sin A at F ~ sin A 


sinB sin Tey 
And because P is always very nearly equal to c,‘the sine of 
a+ pwill differ extremely little from sin iS + c,) and may 
R Sik A” 


sin (ap) F 7 

Hence we manifestly have 
_d. sin a.sin (p-+-p)—d. sin p . . SID Ue: 
“neal R. sin A 
: Which, by taking the expanded expressions, for sin ic + P)s 
and sin (asc? ) and reducing to seconds, gives 

d sin P. sin (ap) 
sin 1” R.SIN A " 

3. When either of the distances Ry Ly becontis’ infinite, 
with respect to d, the corresponding term in the expression 
art. 1 of this problem, vanishes, and we have accordingly 

fee d. sin Lig ee a d .sin (PB) 

~ BS 8ipe tk * R. sin 1” 

The first of these will apply when the object ais a heavenly 
‘body, the second whens is one. When pour A and Bare 
such, then c—p=0. 


~ But without supposing either a or B infinite, we may have 
¢— p= 0,orc = Pin innumerable instances: that is, in all 
cases in which the centre P of the instrument is placed in the 
circumference of the circle that passes through the three 
points 4,8, c; or when the angle sec is equal to the angle 
BAC, or to Bac+180°. Whence, though c should be inacces- 
sible, the angle acs may commonly be. obtained by observa- 
tion, without any computation. It may further be observed, 
that when rp falls in. the circumference, of the circle passing 
through the three points a, B, c, the angles a, 8, c, may be 
determined solely by measuring the angles - APB ate Bec. For, 
the opposite angles asc, apc, of the quadrangle inscribed in 
a cirele, are (theor. 54 Geom. ) = 180°. Consequently, azc 
=180°—arc, and Bac BE LGD es anor Ae) =180° — net: 
APB). 


4, If one of the nia viewed from a further sition: be 
a vane or staff in the centre of a steeple, it will frequently 
happen that such object, when the. observer comes near it, is 
both invisible and inaccessible. Still there are various me- 
thods of finding the exact angle at c. Suppose, for example, 
the signal-staff be inthe centre of a circular tower, and that 
“the angle APB was taken at p-near its base. Let the tangents 
eT, PT, be marked, and on them two equal and arbitrary dis- 

Vou. IT. if tances 


But the triangle ABC gives L = 


therefore be substituted for it, marine L=-—-; 








CR 


74 TRIGONOMETRICAL SURVEYING, 


tances pm, Pm’, be measured. Bisect 
mm at the point x; and placing there a 
‘signal-staff, measure the angle pz, 
which, (since rn’ prolonged obviously 
passes through c the centre,) will be 
‘the angle p of the preceding investiga- : 
tion. Also, the distance ps added to the radius cs of ie tows. 
er, will give pc=d in the former investigation. 

If the circumference of the tower cannot be measured, and 
the radius thence inferred, proceed thus : Measure the angles 
Beg, ppt’, then will Bpc=3(BPT-+-BPT’)—=p ; and cpT—=sBeT— 
BPC: Measure pr, then pc = pT. sec cpr=d. With the va- 
lues of p and d, thus obtained, proceed as before. 

5. If the base of the tower be polygonal and devaliie as 
“most commonly happens ; assume P in the point of intersec- 
tion of os of the sides prolonged, - 
and ppc =i (get.-+ ppt’) as before, 
pr= the distance from r to the mid- 
dle of one of the sides whose pro- 
longation passes through Pp; and 
hence rc, is found, as above. If the 
figure be a regular hexagon, then 
the triangle rmm’, is equilateral, 

aod PC=m mf. , 








FROGEEM Hil. 3 


Po Reduce ih les measared in a Plane fnetined to the Hori- 
zon, to the Cor responding Angles i in the Horizontal Plane. 


Let sca be an angle méasured in’a plane. jnclined to the 
horizon, and let w’ca’ be the corresponding angle inthe ho- 
rizontal plane. Let'd and d’ be the zenith ‘distances, or the 
complements of the angles of elevation aca’, Bex’. Pies 
from z the zenith of the observer, , | 
or of the angle c, draw the arcs za, ; 
zb, of vertical circles, measuring the 
zenith distances d, d, and draw the | ; 
are aé of another great circle toi) 
measure the angle c. It follows § : 
from this construction, that the an- | 
gle z, of the spherical triangle zab, 
is equal to the horizontal.angle a'cB’ 7 ‘and that, to find it, the 
three side za=d,; zb=d’, ab=c, are given. Call the sum of - 
these ¢; then the resulting formule of prok, 2 ch. iv, appl 
ed to the present instance, becomes - 





ain 


pale 


GEODESIC OPERATIONS, | 28 


sin 7(s—d) . sin ¢ ing (Gd) 
sin “ad. sin ag | 
if hand h’ represent the angles of aititude ACA’ BOB’, _ the 
preceding expression will become | 
— ysini(e-+h—h) . sind (e gi mio 
sae Le cosh . cosh’ 
Or, in logarithms, 
log sin } kc = 4(20 a log sind (c $A—h') af a sin. 
i(c¢ fe ir ~h)'— log cosh — log cosh). 
Cor. 1. If h =X, then is'sin 1 es _ 
log sin La’cn = 10 + log sin acne toe cos h. 
Cor. 2. If the angles h and hi be: very small, and nearly 
equal; then, since the cosines of small angles vary SuEmely 
slowly, we may, without sensible error, take : 
log sin 3a‘cs' = 10 + log sin Lacs—log cos 4 (h--h). 
~ Cor. 3. In this case the correction cA ‘op —acB, may be 
found by the expression 
eha. 


x = sin 1” (tan 2 1c Os Sie Nee 


sin dessin 5c= of —— 


: and 








3 —cot 2 aie 5), 


And in this formula, as well as the first given for sin rae d-and 
d’may be either one or both greater or less than a ‘quadrant ; 
that is, the equations will obtain whether aca’ and scp’ be each 
an elevation or a depression. 

Scholium. By means of this problem, if the altitude of a 
hill be found barometrically, according to the method describ- 
ed in the ist volume or geometrically according to some of 
those described in heights and distances, or that given in the 
following problem ; then, finding the angles formed at the 
place of observation, by any objects in the country below, and 
their respective angles of depression, their horizontal angles, 
and thence ‘their distances may be found, and their relative 
places fixed in a map of the country ; taking care to have a 
sufficient number of angles perpese intersecting lines, to very 
the ape raiont. 


PROBLEM iV. 


Give the Anplés of Elevation of Any Distant cbfeet, taken 
at Three places in a Horizontal Right Line, which dves not 
pass through the point directly below the object ; and the 
i Reanective Distances between the stations ; to find the 
Height of the Object, and its Distance from either station. 


Let arp be the horizontal plane ¢ rez the perpendicular 
height of the object F above that plane; a, 8, c, the three 
places of observation ; ; FAE, FBR, FOE, the respective angles 

. of 


76. TRIGONOMETRICAL SURVEYING. 


of elevation, and as, sc, the giv- 
en distances.. Then, since the | 
triangles anF, BEF, crF, are all 
right angled at 2, the distances. 
AE, BE, cE, will manifestly be as - 
the cotangents ofthe angles of. 
elevation at 4,8, and.c: and we . 
have to determine the pointe, — ee *) NCD 
so that those lines may have that - 

_ ratio. To effect this geometrically use the following 

Construction. ‘Take pM, -on ac produced, equal to BC, BN 

equal to an ; and make | 
MG : BM (= BC) 7: cota: cot B, 
and BN (= 4B): NG 3; cot B: cotc. 
With the lines my, mo, ne, constitute the triangle mnc.; and 
join sc. Draw at so, that the angle raB may be equal to mee ; 
this line will meet sc ‘produced i in £, the point in the horizon- 
tal plane falling perpendicularly below F. 
Demonstration. BY the lene triangles AEB, GMB, We 
have AE : BE ::.MG : MB fia A: cot B, 
and BE: BA on BN) :: BM : BG. 
Therefore the Arip les BEC, BGN, are ‘similar ; ; consequently 
BE 2 EC :: BN:NG:: cotB:cotc. Whence it is obvious that 
AE, BE, CE, are respectively. as cot a, cot B, cote. 

Calculation. In the triangle men, ‘all the sides are given, to 
find the angle cun = angle azz. Then, in the triangle mcs, 
two sides and the included angle are given, to find the angle 
MGB = angle ras. Hence, in the triangle aes, are known aB 
and all the angles, to find az, and ¥ BE. And then EF = aE. tan 
A= BE. tans, : tains 





Otherwise, , independent of the construction, thus, 


Put 4B =p, ec o=d, EF =2; and then. express algebrai- 
~ cally the followi ing theorem, given: at p. 128 Sains s Select 
Exercises : 
AE? , BC + CE? ..AB = BE? . AC fo AC. AB. BC, 

the line ex being drawn from the vertex z of the triangle ACE, | 
to any point B inthe base. ‘I'he equation thence originating is 
da? . cot? a-+-nz? . cot? c = (p+-d)xz2 . cot? B+ (p+d) pd. 
And from this, by transposing all the unknown terms to one 
side, and extracting the root, their results 


na (p+d) pd 
Shor sph Ge nad d . col? wD cot? ¢—(p+d) cag : 


Whence 


-GEODESIC OPERATIONS. 17 


‘Whence sr is known, and the distances AL, BE, cE, are readily 
founds. 
Cor. When p=d, orp + d= 2p =2d, she expression be- 
comes better suited for logarithmic computation, being then 
a=d—,/ (4. cot? a+} cot? c—cot® B 
In this case, therefore, the rule is as follows ; Double the log. 
cotangents of the angles of elevation of the extreme stations, 
find the natural numbers answering thereto, and take half their 
sum; from which subtract the natural number answering to 
twice. the log. cotangent of the middle angle of elevation: 
then half the jog. of ‘this remainder subtracted from the’ log. 
of the measured distance between the Ist and 2d, or the 2d, 
and 3d stations, will be the log. of the height of the object. 


- PROBLEM V. 


In any Spherical Triangle, knowing. Two Sides and the In- 
cluded Angle ; it is required to find the Angle Comprehend- 
ed by the Chords of those two sides. 


Let the angles of the spherical tri- 
angle be a, B, c, the corresponding an- 
gles included by the chords a’, B’, c’ ;_ 
the spherical sides opposite the former 
a, 6, ¢, the chords respectively Oppo- 
site the latter «, B, y; “then, there are 
given 6, c, and a, to find a’. | 

Here, from prob. 1 equa. chap. iv, we have 

cos a=sin b . sinc . cos fae b.cos ¢. 

But cos ¢ = cos (Le + 1c) = cos? 1c—sin? Le ‘(by equa. v 
ch. iii) = (1—sin? }c) —sin® de==1—2sin? dc. And in like 
manner cos a= 1—2 sin? 1c, and cos 6b = 1— 2 sin? 2b. 
Therefore the preceding equation becomes | 
1—2 sin? 1a =4 sin 16.cos 16.sinic.cos} Ze + COS A+ 

(1—2 sin? 30) (F=12 sin? Ae), 
But sin ta=-1e, sin 1b=46, sin }e=1y: which values sub- 
stituted in ina equation, we obtain, after a little reduction, 











2 Aye nme ceZ 
oxP TE = by . cos fea . cos Le. cos fg Be 182 y?2, 
py) Bsa gg 2 x 
Now, (equa. 11 ch, iii), Gas) see ma vith _ Therefore, by 


substitution, 
By . cos a=By . cos 40. cos dc . cos a-+} a8? ys 
whence, dividing BY BY; ‘there results 


cos a’=1 b. cos de / cos a+18 hy 
or, lastly. by restoring the values of ie, 4%; we: eres: 
cos a'=cos 14. cos de. cosa-sin 1h. sin eo... (L.) 


me TRIGONOMETRICAL SURVEYING. 


Cor, 1... It follows evidently from this formula, that when 
the spherical angle is right or obtuse, it is always greater than 
the corresponding angle of the chords. 

. Cor. 2. The spherical angle, if acute, is less than the ¢ cOor-_ 

“responding angle of the chords, cnnee we a Haye COS A greater 
aes than sin 16. sin t¢ : 

mt J—cos 1b. sin de. 

PROBLEM VI. 
iuovine Two Sides and the Included Angle of a Rectilinear 
Triangle, it is required to find the Spherical Ee of the 
"Two Arcs of which those two sides are the chords. 


Here g, v, and the angle a’ are given, to find a. Now, 
since in all cases, cos = ,/(1~sin?), we have | 

cos 1b. cos lo=/[(1- ~sin? 4b). (1—sin? 3¢)] ; i 
we have also, as above, sin 1b=18, and sinic=ly. Sub- 
stituting these: values in the equation 1 of the preceding ae 
lem, there will result, by reduction, 

cos A‘ — TRY (II. 

J (— $8). (1448). (ty) hy) 

To compute by this formula, the values of the sides 2; Ys 
must be reduced-to the corresponding values of the chords of 
a circle whose radius is unity. This is easily effected by di- 
viding the values of the sides given in feet, or toises, &c. by 
such a power of 10, that neither of the sides shall exceed 2, 
the value of the greatest chord, when radius is equal to unity. 

From this investigation, and that of the preceding prone, 
the following corollaries may be drawn. 

Cor, 1. If c=b, and of ‘consequence y = 8, ‘then will 

- COs A*==c0s A. cos? 1c-b sin? 1¢; and thence 

1—2 sin? $a’ = (1-2 sin? La) cos? ¢ + (1—cos? 4c): 
from which may be reduced 
/o. sin da’sssin La. cos 2¢. 2. III, ) 
“on. Be Also, since cos: jo=,/(1—sin? 40) =4/(1— 37? » 
equa. 1 will, in this case, reduce to 
gin La’, 

Zia ape 8 
i ON. Ss “ior the equation 11, it appears that the verti- 
cal angle of an isosceles spherical triangle, is always greater 
than the corresponding angle of the chords. 

Cor. 4..1f A==90°, the ee I, it; Bre 
cos A’==sin ie sin 4 =1py . wey. yo - 
These five formule are Pig and rigorous, whatever, he 
the magnitude of the triangle. But if the triangles be small, 
ihe ares may be put instead of the sines in equa. v, then 

Yor. 5. As cos a'=sin (208 ml wad a = in this case, 90°—a’; 
the 





cos A == 


’ 1 
_ sin la= 


ie 


1 \4 


GEODESIC OPERATIONS. "9 


the small excess of the spherical right angle over the corres-_ 
ponding rectilinear angle, will, supposing the arcs 6, c, taken 
in seconds, be given in seconds by the following expression 
isi the ¥ be | 
aia 90° —a’= * ace My Pu . (VI. ) ) 

The error in this formula will not amount toa second, when 
b-+c is less than 10°, or than 700 miles measured on the 
earth’s surface. 

Cor. 6. If the hypothenuse does not exceed 11°, we may 
substitute a sin c instead of ¢, and a cos c instead ae 5; this 
will give dc=a? . sinc. cos c. =}a? .sin 2 lips et 
sin 28 ; whence 

) : oe a? .sin2c a? . sin a8 
: (90° —s)= ~ SR” es: 8R . (VI yes 
If a=119°, and p==c=45° nearly ; then will 90°—a’=17".7 

Cor. 7. Retaining the same 2 hypothesis of tint and a= 
or < 142, we have 

, 62 cot Bbc 


B=—B = 4 BRE. Ba ee ee (VII) . 


Also i Cae ae, y (IX.) ; 








Cor. 8. Comparing formule vit, 1x, with v1, we have pr’ 
=o—c’==1 (909—a’.) Whence it. appears that the sum of 
the two excesdes of-the oblique spherical angles, over the 
corresponding angles of the chords, in a small right-angled tri- 
angle, is equal to the excess of the right angle over the cor-— 
responding angle of the chords. So that either of the for- 
mule v1, vit, vir, 1x, will suffice to determine the difference 
of each of the three angles of asmall right-angled spherical 
triangle, from the corresponding angles of the chords, And 
hence this method may be applied to the measuring an arc of 
the meridian by means of a series of ook See arts. 8, 
9, sect. 1 of this chapter. | | 


PROBLEM VII. 


Ina Spherical T tiansle asc, Right Angled in a, eee the 
_Hypothenuse sc (less than 4°) and the Angle », it is requir- 
ed to find the Error e committed through finding by Plane 
Trigonometry, the Opposite Side ac. 


Referring still to the diagram of prob. 5, where we now. 
suppose the spherical angle a to be right, we have (theer. 10 


~ chap. iv) sin b=sin a. sins. But ithas been remarked at pa. 


381 vol.i, that the sine of any arc is equal to het: sum of 
the following sefies ; 


Sin - 


80 TRIGONOMETRICAL SURVEYING. 


RB PE ES CS At 


eae “3 sin o eae ge es Ae FRR ee aT Ue 
ay 53° 0346 2345.07" 
3 5° ; qo 
or, sin aza— = Beer wa Ras AS —— -+- &c 


6 120 ~- 5040 
‘And, in the present enquiry, all the terms after the second 
may be neglected, because the 5th power of an arc of 4° di- 
vided by 120, gives a quotient not ExCLenIng 0”. 01. Conse- 
quently, we may assume sin 6 = 6-13, sina=a—1a*; and 
thus the preceding equation will become, | 
| b—1b?= sins (a—a?) 
: or b=a sin B—i (a> . sinB—b?), 

Now, if the triangle were considered as rectilinear, we 
should have b=a . sin 8; a theorem which manifestly gives 
the side 6 or ac too great by 1 (a?.. sins -- 5b), But, neg- 
lecting quantities of the fifth order, for the reason already as- 
signed, the last equation but one gives 6° =a? . sin? B. There- 


fore, by substitution, e= — 1 a3 . sin B (1— sin? B): or, to 
have this error in seconds, take R= the radius expressed in 
a? . cos? B 
seconds, so shall e= —a. sins . —{——— 
: Gr’ Rr” 
Cor. 1. If a=4°, and 6==35° 16’, in which case the value 
of sin . cos? B is a maximum; we shall finde = —41”. 


Cor. 2. If, with the same data, the correction be ‘applied, 
to find the side c agjncen to the given angle, we should have 
Grirs ‘sind B 
e'=a . cos B taign | 
So that this error exists in a contrary sense to the other; the 
one being subtractive, the other additive. . 
Cor, 3. The data being the same, if we have to find the 


angle C, the error ue be corrected will be 


pe os Sto RR 

=O" AR? ‘ 
As to the exvess of the arc over its chord, i is , easy to find it 
correctly from the expressions in prob. 5: but for arcs’ that 


are very small, compared with the eiafiias a near approxima-. 
tion to that excess will be found in the same measures as the 
radius of the earth, by taking ;', of the quotient of the cube 
of the length of the arc divided by the square of the radius. 


PROBLEM VIII. 
itis required to Investigate a Theorem, by means of which, 
Spherical Triangles, whose Sides are Small compared with 
the radius, may be solved by the rules for Plane T rigono- 
metry, without considering the Chords of the Bee tira 
Arcs or Sides. as 
Let 


SPHERICAL EXCESS. 81 


Let a, b, c, be the sides, and a, 8, c, the angles of a sphe- 
rical triangle, on the surface of a sphere whose radius is r ; 
then asimilar triangle on the surface of a sphere whose radius 


Folie b 
== 1, will have for its sides <, yt : ; which, for the sake of 


brevity, we represent by «, £, y, respectively : then, by equa. 
COS #—cos B .cos y 
sin 8. sin y 
Now, r being very great with respect to the sides, a, b,-c, 
we may, as in the investigation of the last problem, omit all 
the terms containing higher than 4th powers, in the series 
for the sine and cosine of an arc, given at pa. 381 vol. i: so 
shall we have, without perceptible error, 
diane a? a4 . Bs 
me aT e34 " 
And similar expressions may be adopted for cos B, cos 7, 
siny. Thus, the preceding equation will become 
con a wb (Stch ota!) hte (ot Bt mat) het 
py (1—3 B?—ty? 
Multiplying both terms of this fraction by 1+} (@2 + y2), to 
simplify the denominator, and reducing, there will result, 
B2+y2— wat +Br—yt —2a2p? —Qery?2 28272 
€08 ass 4 ° 
2By ube 24Ry 
Here, restoring the values of «, 8, 7, the second member of 
the equation will be entirely constituted of like combinations 
of the letters, and therefore the whole may be represented by 
M N 
cos A = oF bY ogpive’ eu Ch) 
Let, now, a’ represent the angle opposite to the side a in the 
the rectilinear triangle whose sides are equal in length to the. 
arcs a, 6,c; and we shall have 
62 +c? —a? M 
Phe se She | 
Squaring this, and substituting for cos? a its value 1— sin? a‘, 
there will result 
—462 c2 sin? a’ =a2-+b2 +c? — 2a? b? —2a? c2 —2h% c2@ = y, 
So that, equa. 1, reduces to the form | 


1, chap. iv, we have cos a = 





cos A = 


‘ 


PES a eta 
COS A == cos A’ ——-— SIR? A. 
6r2 


Let a = 4’+ x, then, as x is necessarily very small, its second 
power may be rejected, and we may assume cos a = cos a-— 
x. sin a’; whence, substituting for cos a this value of it, we 


shall have x = oe) sin A. 
672 
Von. WH. 12 It 


ge TRIGONOMETRICAL SURVEYING. 

- It hence appears that x is of the second order, with respect 

to” and = ; and of course that the result is exact to quan- 

faties within the fourth order. Therefore, because a==a'--z, 
| A=A +o. sin a’, 


But, by prob. 2 rale 2, Mensuration of Planes tbc sin ‘i is the 
area of the rectilinear triangle, whose sides are a, 6, andc. 





area 
Therefore 4 = a + —_: 
: 3r2? 
Ora =a Binds 5 
3r2° ; 
in ike (yg 
manner Sr? 
i area 
or 
on aah Pe) bE: area 
And a’-- 8B -+ c’ = 1802 Pe Ach Be oo ce 
area thie w fies 
r, ——~ = a+ B+¢—180°. 


rs 
Whence, since the spherical excess 1s a measure of the area 
(th. 5 ch. iv), we have this theorem: viz. 

Al spherical triangle being proposed, of which the sides are 
very small, compared with the radius of the sphere ; if from 
each of its angles one third of the excess of the sum of tts three 
angles above two right ungles be subtracted, the angles so dimi- 
nished may be taken for the angles of a rectilinear triangle, whose 
side are equal in length to those of the proposed spherical trian- 

le.” | 

‘ Mi Scholium. : 

We have already given, at th. 5 chap..iv, expressions for 
finding the spherical excess, in the two cases, where two sides 
and the included angle of a triangle are known, and where 
the three sides are known. A few additional rules may with 
propriety be presented here. 

1. The spherical excess E, may be found in seconds, by the 


- RS ee : fa vas 
expression rE = ——; wheres is the surface of the triangle 
T qe : ‘ sf - 


i er ; SiInB. sine 
=1be. sina =1ab. sinc ==1ac . sin B = ta? . -_____, 
ae ee Bid ; sin (B--c) 





This curious theorem was first announced by M. Legendre, in the Memoirs of’ 
the Paris Academy, for 1787. Legendre’s investigation 1s nearly the same as the 
above; ashorter investigation is given by Swanberg,‘at pa. 40, of his “ Exposition. 
des Opérations faites en Lapponie ;” but it is defective in point of perepieny 

: . £ IS 


, 


SPHERICAL EXCESS. 83 


ris the radius of the earth, in the same measures as a, b, and c, 

and n” == 2062648, the:seconds in an arc equal in length to 
the radius. ~ . sas 

If this formula be applied logarithmetically ; then log x’ = 

1 . 

.——, = 5:3144251, 
log arc 1’ 2 


2, From the logarithm of the area of the triangle, taken 
as a plane one, in feet, subtract the constant log. 9-3267737 
then the remainder is the logarithm of the excess above 180° 
in seconds nearly*. | 

3. Since s = }be . sin a, we shall manifestly have 5 = 


R” i ; 
=; bc. sin a. Hence, if from the vertical angle B we demit 


= 


the perpendicular sp upon the base ac, dividing it into the 
two segments #, 6, we shall have b=«-+,, 
R" : Wi 
and thence gk = Spa (#48) sin Ame on, ae. 
“ 


y R : : 
gin A A Pe fc.sin a. But the two right- 





angled triangles agp, cep, being nearly 
rectilinear, give «=a@.cosc,andp=c. 


60s A; whence we have 
u da 


R f : 4 
E==—— dc.smA.cosc —— ¢? , gina. cos A. 
Qr2 fot Or2 


In like manner, the triangle anc, which itself is so small as te 
differ but little from a plane triangle, gives c . sin az=a.. sinc. 
Also, sin a. cos a==}sin 2a, and sinc . cos c=} sin 2c 
(equa. Xv. ch. iii). Therefore, finally, : 


: R ; 
E == ——a?.. sin 2c + ——c? . sin 2a. 
4r? 412 


From this theorem a table may be formed, from which the 
spherical excess may be found; entering the table with each 
of the sides above the base and its adjacent angle, as argu- 
ments. 
4. If the base b and height h, of the triangle are given, 


then we have evidently & = tbh Hence results the fol- 


Jowing simple logarithmic rule : Add the logarithm of the base 
of the triangle, taken in feet, to the logarithm of the perpen- 
dicular, taken in the same measure ; deduct from the sum the 
logarithm 9-6278037 ; the remainder will be the common lo- 
garithm of the spherical excess in seconds and decimals. 





* This is General Roy’s rule given in the Philosophical Transactions, for 1790, 


pa. 171. 
' 5. Lastly, 


84 TRIGONOMETRICAL SURVEYING. 


. 5. Lastly, when the three sides of the triangle are given in 
feet; add to the logarithm of half their sum, the logs. of the 
three differences of those sides and that half sum, divide the 
total of these 4 logs, by 2, and from the quotient subtract the 
log. 9°3267737 ; the remainder will be the logarithm of the 
spherical excess in seconds &c. as before. 

One or other of these rules will apply to all cases in whieh 
the spherical excess will be required. 


a 


PROBLEM IX. 


Given the Measure of a Base on any Elevated Level ; to find 
its Measure when Reduced to the Level of the Sea. 


Let r represent the radius of the earth, or the distance 
from its centre to the surface of the sea, r-+h the radius re- 
ferred to the level of the base measured, the altitude h being 
determined by the rule for the measurement of such altitudes 
by the barometer and thermometer, (in this volume) ; let 8 
be the length of the base measured at the elevation fh, and 6 
that of the. base referred to the level of the sea. 
Then because the measured base is all along re- 
duced to the horizontal plane, the two, B and 4, 
will be concentric and similar arcs, to the re- 
- spective radii r-+h andr. Therefore, since si- 
milar arcs, whether of spheres or spheroids, are 
as their radii of curvature, we have 


rB 
h:r::38:b6=>—— 

aa Gee: r-+h 

Hence, also n—b = 8 ~ Ser ; or, by actually dividing 

Bh by rth, we shall have | 





Which is an accurate expression for the excess : of B above b. 

But the mean radius of the earth being more than 21 mil- 
lion feet, if % the difference of level were 50 feet, the second 
and ali succeeding terms of the series could never exceed the 
fraction ;zsccdsa0507 ; and may therefore safely be neglect- 
ed: so that for all practical purposes we may assume B—6 


h 
= —. Or, in logarithms, add the logarithm of the measured 


base in feet, to the logarithm of its height above the level of 
the sea, subtract from thé’ sum ‘the logarithm 7°3223947, the 
remainder will be the logarithm of a number, which taken 
from bi measured base will leave the reduced base required. 
PROBLEM 


GEODESIC OPERATIONS. 35 


_ PROBLEM X.. 


To determine the Horizontal Refraction. 


1. Particles of light, in passing from any object through 
the atmosphere, or part of it, to the eye, do not proceed in 
a right line; but the atmosphere being composed of an infini- 
tute of strata (if we may so call thein) whose density increases 
as they are posited nearer the earth, the luminous rays which 
pass through it are acted on as if they passed successively 
through media of increasing density, and are therefore in- 
flected more and more towards the earth as the density aug- 
ments. In consequence of this it is, that rays from objects, 
whether celestial or terrestrial, proceed in curyes which are 
concave towards the earth ; and thus it happens, since the eye 
always refers the place of objects to the direction in which the 
rays reach the eye, that is, to the direction of the tangent to 
the curve at that point, that the apparent, or observed eleva- 
tions of objects, are always greater than the true ones. ‘The 
difference of these elevations, which is, in fact, the effect of 
refraction, is, for the sake of brevity, called refraction: and it 
is distinguished into two kinds, horizontal or terrestrial refrac- 
tion, being that which affects the altitudes of hills, towers, 
and other objects on the earth’s surface ; and astronomical re- 
fraction, or that which is observed with regard to the altitudes 
of the heavenly bodies. Refraction is found to vary with the 
state of the atmosphere, in regard to heat or cold, humidity 
or dryness, &c, : so that, determinations obtained for one state 
of the atmosphere, will not answer correctly for another, 
without modification. ‘Tables commonly exhibit the refraction 
at different altitudes, for some assumed mean state. \ 

2. With regard to the herezontal refraction the following 
method of determining it has been successfully practised in 
the English Trigonometrical Survey. 

Let a, a’, be two elevated stations on 
the surface of the earth. Bp the inter- 
cepted arc of the earth’s surface, c the 
earth’s centre, au, 4H, the horizontal . 
lines at a, a’, produced to meet the op- 
posite vertical lines cH, cH. Leta,a’, 
represent the apparent places of the 
objects a, a’, then is a’aa’ the refraction Cc 
observed at a, and aa’a the refraction observed at a’; and half 
the sum of those angles will be the horizontal refraction, if 
we assume it equal at each station. Mee 

Now, an instrument, being placed at each of the stations a, 
a’, the reciprocal observations are made at the same instant 

yo of 





86 TRIGONOMETRICAL SURVEYING. 


of time, which is determined by means of signals or watches 


“previously regulated for that parpose ; ; that i is, the observer 


at a takes the apparent depression of 4’, at the same moment 
that the other observer takes the apparent depression of ‘a. 

In the quadrilateral aca’'l, the two angles a, a’, are right 
angles, and therefore the angles 1 and c are together equal to 
two right angles: but the three angles of the triangle TAS 
are together equal to two right angles ; and consequently the 

angles a and a’ are together. equal to the angle ¢c, which 1s 
measured by the arc ppv. If therefore the sum of the two 
depressions na’a, Hao’ be taken from the sum of the angles 
Ha‘aw aa’ or, which is equivalent, from the angle ¢ (which, is 
known, because its measare Bp is known) ; the remainder is 
the sum of both refractions, or angles aa’a, a’sa”%. Hence this 
rule, take the sum of the two depressions froin the measure of the 
intercepted terrestrial arc, half the remainder ts the refraction. 
'. 3. If, by reason of the minuteness of the contained arc gn, 
one of the objects, instead of being depressed, appears ele- 
vated, as suppose a’ toa’: then the sum of the angles a/’ aa’ 
and aa'a will be greater thaw the sum 1aa’-F14’a, or than c, by 
the angle of elevation a’ aa’; buat if from the former sum there 
be taken the depression aa’a, there will remain the sum of 
the two refractions. So that im this case the rule becomes as 
follows: take the depression from the sum of the contained arc 
and elevation, half the remainder is the refraction. 

5. The quantity of this terrestrial refraction is estimated 
by Dr. Maskelyne at one-tenth of the distance of the object 
observed expressed in degrees of a great circle. So, if the 
distance be 10000 fathoms, its 10th part 1000 fathoms, is the 66 
part of a degree of a great circle on the earth, or 1’, which 
therefore is the refraction | in the altitude of the object at that 
distance. 

But M. Legendre is mduced, he says, by several experi- 


‘ments, to allow only ;th part of the distance for the refrac-. 


tion in altitude. So ‘that, on the distance of 10000 fathoms 
the 14th part of which is 714 fathoms, he allows only 44” of 
terrestrial refraction, so many being contained in the 714 fa- 


‘thoms. See his Memoir ne the Trigonometrical ope- 


vations, &c. 
Again, M. Delambre, an ingenious French astronomer, 


- makes the quantity of the terrestrial refraction to be the ith 


art of the arch of distance. But the English measurers, 
especially Col. Mudge, from a multitude of exact observa. 
tions, determine the quantity of the medium refraction, to be 
the 19th part of the said distance. _ ‘ 
‘The quantity of this refraction, however, is found to vary 
€0n,- 


REFRACTION, oF 


considerably, with the different states of the weather and at- 
mosphere, from the 4th to the ;),th part of the contained arc. 
See pe de ait Survey, vol. 1 pa, 160, 355. 


Scholisins: 


Having given the mean results of observations on the ter- 
restrial refraction. it may not be amiss, though we cannot en- 
ter at large into the investigation, to present here a correct 
table of mean astronomical refractions. ‘The table which has 
been most commonly given in books of astrovomy is Dr. Brad- 
ley’s, computed from the rule r==57’X cot (a, + 3r), where 
ais the altitude, r the refraction, and r==2’3 35" when a==202, 
But it has been found by numerous observations that the re- 
fractions thus computed are rather too small.—Laplace, in his 
Mecanique Celeste (tome iv pa. 27) deduces a formula which 
is strictly similar to Bradley’s ; for it is rm X tan (z — nr), 
where z is the zenith distance, and m and m are two constant 
quantities to be determined from observation. The only ad- 
vantage of the formula given by the French philosopher, over 
_ that given hy the English astronomer, is that Laplace and his 
colleagues have found more correct coeflicients than Bradley 
had.) 

s Now, if p=57° 2957795, the arc nil to the radius, if we 


Ar 
make m = i (where kisa constant coefficient which, as well 
as n, is an abstract number ,) the preceding. equation will be- 
mr 
come — = kX tan (z—nr). Here, as the refraction 7 is ale 
ways very small, as well as the correction nr, the trigonome- 
trical tangent of the arc nr may be substituted for _ ; thus 


we shall have tan mr==k . tan (2—nr). 


But nr=}2 -(dz—mr)....3~nr=j2+(j2—nr) ; 
t at Snr : ; 
Ce tan ar ne 2 — sing —sin(z—2nr) 
ae tan tan (z—nr) 2, 2—2nr sinz-+sin(z—2nr) 
tan —--———— | 
: 2 | 
Bae. sin aa ae sin Zz. 





—k. 
This formula i is easy to use, ; ry i 
are known: and it has been ascertained, ie a mean of many 
observations, that these are 4 and 99764175 respectively, 
‘Thus Laplace’s equation becomes 


sig 


88 TRIGONOMETRICAL SURVEYING. 


7 sin (¢—~8r) = == °99765175 sin z: 
inl from this the following table has been computed. Besides 
the refractions, the differences of refraction, for every 10 
minutes of altitude, are given; an addition which will render 
the table more extensively useful in all cases where great ac- 
curacy is required. 


Table of Rifribibons. . 
Borers 29 92 inch. Fah. Thermom. 54°. 
































Alt . Diff. -), Alt. Diff. |} Alt, Diff.;, Alt. Dif 
Japp. Reftae | on 10 ap. Retr 107. app. 10’. ee 
.mM.} M.S. S.. il, M.{ M. 8..} 8. D. |M. 8S. s. S. “0:95 
0\33. 46°3| 112.0/17 O17 24°81. 9°5 |}. 14 |3 49°8 |2-58 39°3) 0:24 
10/31 543} 105°0|| L017 15°83! 9°O |} 15 |3 343 \2-28 37°8| 0°24 
20|30 9°3| 97°3| 20/7 6'3| 86 || 16 |3 20°6 |2°02 36°4| 0-23 
30/28 321} 89°8| 30/6 57°7| BL |}. 17. |3 °° 85 1°82 350} 0:22 
40/27 2:2] 83°6}| 4016 49°6) 7°7 || 18 |2 57°6/1°65 336; 0°22 
- §0/24 386} 77'4)| 50/6 41°9| 7°5 || 19 12 47°7 [1°48! 32°3} 0-21 
0/24 21-2! 75°68. O16 3e4), 7°3 |] 20 |2 38°8 11°37 31°0| 0°21 
10/23 9°6) 662)! 10/6 2771] 71 || 21 |2 30°6 /1-24, 29°7|} 0°20 
20/22 34) G61°5|| 20/6 20°0, 6°9 || 22 [2 23°2/1-11| 28°4) 0°20 
30)21 4°9| 67°1]) 30)6 131] 67 |) 23 [2 16°5|1-05) 27°2| 0-20 
40)20 4°8) 53°3}}' 40/6 64) 6°5 |) 24 |2 10°2/0'98: 25°9; 0°20 
50\19 11°5| 49'3|| 50/5 59°9} 63 || 25 |2 4310-901] 67. |24-7| 0-20 
O|18 222) 45°9))9 O|5 53°6| 6°2 || 26 |1 58°9 0°83) 23°5| 0°20 
10/17 36:3} 43°1!] 10/5 47°41 -5°9 || 27 |1 53°9 1078 29°4| 0°20 
20/16 53°2/ 39°8)}| 20/5 41°56) 5°7 || 28 {i 49°2 (0-73) 21°21 0-19 
30116 13-4) 37°4/| S05 35°8| 5°5 || 29 |1 44°8'0°70 20°0| 0-18 
40/15 36:0] (35°1|} 40/5 30°3) 5°3 || 30 {1 40°6 |0°65' 18-9] 0-18 
~50)15 0-9}  32°8\| -50/5 25:0] 5°2 || 31 {1 36°7 |0-60) 7-8; O18 | 
O}14 28-4] 30°8)}10 O15 19°83] SL |} 32 |1 33°1\058 16°7|- 0-18 
10}13 57-3] 28°8|| 10)5 147] 5°O || 33 {1 29°6 10°56) 15°6| O17 
20}13 28:5} 27:2) 20/5 97) 48 || 34 |1 262 0°53" 145} O17 
30!13 1:3). 25°7|| 30/5 4°9| 4°6 || 35 |1 2371 0°50 13°5|' O17 
40|L2 35-6} 24:3]! 4015 O°3] 44 |} 36 |1 20°1 0-48 12:4) 0-17 
50/12 11:3} 23°0]| 50/4 55°9| 42 |} 37 |1 17°2'0-47 14:3] 0-17 
O}11 483) 2171/11 0/4 51°7| 41 |} 38 {1 14°4/0-43 10:3) 0-17 
10}11, 26-6} 20°51). 10)4 47°6} 4:0. |] 39 IL 11°8\0-42 92} 0-17 
20/11 61} 194)| 20/4 43°6| 4:0 |} 40 |i 9°3 (0°40 8-2] O-17 
30|10 46:7) 18-44) 30/4 596} 3:9 |] 41 |t 69 10-38! 7:21 0-17 
40|10 28-3] 17:4]| 40/4 35°7] 3:9 |} 42 |1 460-37 6-1] O17 
50)10 10-9] .16°6]} 50/4 31°8} 3°8 || 43 |L  2°410°35 5-1] O47 
0} 9 543} 15°9/|12 O]4 28°0] 3°7 || 44 11 0310-54 &1} O17 
tO} 9 384] 15-0|} 10/4 24°31 $°6 |] 45 10 58°210:33 3:1} 0-17 
201 9 23-4) 14°44) 20)/4 20°7| 3:5 |] 46 [0 56°2/0:32 2-0} 0.17 
30} 9 9-0} 13°7]| 30/4 17°2| 34 || 47 |0 54°310-31 1:0} 0.17 
40] 8 553] 13°0]| 40/4 15°83} 3:2 |} 48 10 52°4/0°30 0-0 
50} 8 42-3} 12-4] 50/4 10°6! 3:1) 49 10 50°6/0-29 
0} 8 29-9}. 11°8/113 O14 7°75) 31 || 50 (0 48°9/0-28 
10} 8 181) 11°5]} 10/4 4°41.3°0 || 51 (0 47°2 10-27 
20} 8 6-6] 11-0] 20/4 1°41 30 |] ‘52 10 45°5 10-26 
— 80} 7 55-6} 10°6)} 30/3584) 29 |} 53 10 43°9 10-26 
40} 745-0} 10-3)| 40/3 55°5| 29 || 54 10 42°3|0,25 
50) 7 347) -9-9]} 5013 52°6] 2:8 || 55 : 40°8 10.25 
7 O| 7 248 14 0}3 49°38 56 |0 59'3 








PROBLEM 


r 


REFRACTION. e9 
- PROBLEM XI. 
‘fo find the Angle made by a given Line with the Meridian. 


1, The easiest method of finding the angular distance of a 
given line from the meridian, is to measure the greatest and 
the least angular distance of the vertical plane in which is the 
star marked a in Ursa minor (commonly called the pole star), 
from the said line: for half the sum of these two measures 
will manifestly be the angle required. 


2. Another method is to observe when the sun is on the 
given line ; to measure the altitude of his centre at that time, 
and correct it for refraction and parallax. © Then, in the sphe- 
rical triangle zps, where z is the zenith Zz 
of the place of observation, Pp the. ele- 

vated pole, and s the centre of the sun, ig 
_ there are supposed given zs the zenith : 
distance, or co-altitude of the sun, rs 
the co-declination of that luminary, Pz 
the co-latitude of the place of observation, and zrs the hour 
angle, measured at the rate of 15° to an hour, to find the an- 
gle szp between the meridian pz and the vertical zs, on which 
the sun is atthe given time. And here, as three sides and one 
angle are known, the required ss is readily found, by say- 
ing, as sine ZS ; sine ZPs :; sine Ps : sine pzs; that is, as the 
cosine of the sun’s altitude, is to the sine of the hour angle 
from noon; so is the cosine of the sun’s declination, to the 
sine of the angle made by the giver vertical and the meri- 
dian. : 


Note. Many other methods are given in books of hy geet 
my ; but the above are sufficient for our present purpose, 
The first is independent of the latitude of the place ; ; the se- 
cond requires it. 


PROBLEM XII. 


To find the latitude of a Place. 


The latitude of a place may be found by observing the 
greatest and least altitude of a circumpolar star, and then apr 
~ plying to each the correction for refraction ; so shall half the 
sum of the altitudes, thus corrected, be the altitude of, the 
pole, or ffs latitude, 

<¥ on. 13 For, 


$0‘ TRIGONOMETRICAL SURVEYING. 


For, if p be the elevated pole, s¢ 
the. circle described by the star, rr 
== £z the latitude: then since ps = 
rt, PR must be = 3 (Ri--Rs).. 

This method is obviously indepen- 
dent of the declination of the star : 
it is therefore most commonly adopt- .__ 
ed in trigonometrical surveys, nm. 
which the telescopes employed are eh tay kM 
of such power as to enable the observer to see stars in the | 
day-time : the pole-star being here also made use of. 

Numerous other methods of solving this problem likewise 
are given in books of Astronomy ; but they need not be de- 
tailed here. feet 4 

Corel. If the mean altitude of a circumpolar star be thus 
measured, at the two extremities of any arc of a meridian, the 
difference of the altitudes will be the measure of that arc : 
and if it be a small arc, one for example not exceeding a de- 
gree of the terrestrial meridian, since such small arcs differ 
extremely little from arcs of the circle of curvature at their 
middle points, we may, by a simple proportion, infer the 
length of a degree whose middle point is the middie of that 
arc. | (ata 





Sch élisitn: 


Though it is not consistent with the purpose of this chap- 
ter to enter largely into the doctrine of astronomical spherical 
problems ; yetit may be here added, for the sake of the young 
student that if a == right ascension, d =declination, | = la- 
titude, a = longitude, p = angle of position (or, the angle at 
a heavealy body formed by: two great circles, one passing 
throvgh the pole of the equator and the other through the 
pole of the ecliptic), = inclination or obliquity of the eclip- 
tic, then:the following equations, most of which are new, ob- 
tain generally, for all the stars and heavenly bodies, 

. fana =tan a .cosz—tanl. sec a. sin iz. 

sin d= sin a. eos 1. sin'7-+sin a. cosié. 

tan A = sin t. tand .seca-+-tan a . cost. 

sin | = sind. cosi—sin a. cos d. sin @. 

. cotan p=cosd.seca.cot¢+sin d. tan a. 

. cotanp = cos 1. sec a. cotz—sin l. tan A. 

. cos a. cos d =cosl.cos a. 

. sin p.cos d= sin 7. cos A. | 

. sin. p . cos A = Sin 2. cos o. 

. tan a= tan A. cos 27. @ when /==0, as is always the ease 
. co$ A= cosa.cosd.§ — with thesun. : 


ye OO LOCO NI OT OT BR tO ND 


eee Se 


The 


| TO FIND THE LATITUDE. ~ 94 


The. inveitgntion of these equations, whickd Is Omitted for 
the sake of brevity, depends on the resolution of the spheri- 
cal triangle whose angles are at the poles of the ecliptic and 
equator, “and the given star, or luminary. 


PROBLEM XIII. 


To determine the Ratio of the Earth’s Axes, and their Actual 
Magnitude, from the Measure of a Degree or Smaller Por- 
tion of a Meridian in Two Given Latitudes ; the earth being 
supposed a spheroid generated by the rotation of an n ellipse 
upon its minor axis. 


Let apBe represent a meridian 
of the earth, pe its minor axis, 
~aB a diameter of the equator, m, 
m, arcs of the same number of de- 
grees, or the same parts ofa de- 
gree, of which the lengths are 
measured, and which are so small, 
compared with the magnitude of 
the earth, that they may be considered as cbindlaing with arcs 
of the osculatory circles af their respective middle points ; let 
Mo, mo, the radii of curvature of those middle points, be=R 
and r respectively ; ; MP, mp, ordinates perpendicular to ag : 
suppose further cp==c, ca=d 3 d? ~c? =e? cp=a2 3 cp=u ; 
the radius or sine total =1 ; the known angle gsm, or the lati- 
tude of the middle point m,= 1; the known angle gsm, or the 
latitude of the point m, =! ; the measured lengths of the arcs 
m and m being denoted by those letters respectively. . 

Now the similar sectors whose arcs are m,wn, and radii of 
‘curvature R,r, give R: 7 >; ™M:m; and consequently RM=rM. 
The central equation He the ellipse hind ay at p. 533 of 


volume first gives PM = ov (d?'—a?) ; pm=— av (d2 ua) ; ; 





-also sp = a ‘pa (iy th. 17 Ellipse). And the method 


of finding the radius oe curvature (Flux. art. 74, 1B); applied 
to the central equations above, gives : 














4 e242 2 ad* —e2 u2 

Ran a” - ‘i ; andr = ( - : i . On the other hand, 
the apiabely SPM gives SP : PM :; COS L : Sin.L; that is, 

a rene | yest dcos?1. 
ae a @ —#?)::cosL: sin iL; whencez eer ee eC 

aia A COS3. 
2S 
And from a Tike process there resuls, oy es wind 


Sub- 


oe TRIGONOMETRICAL SURVEYING. 


Substituting in the equation rm = ry, for R, and r their va- 
lues, for x? and u? their values just found, and observing that 
sin? L + cos? 1 = 1, and sin? / + cos? 1 = 1, we shall find 


(d? —e? saves (d? — e? sin? [2 
3 
or 7A Sot —e*sin? /)? =m (a — e2 sin? L)S. 


or m(da—e? sin? 1) = mw (d? —e? sin? 1). 
_ From this pe picked e? = d? —¢? (by hyp.) = 
d2 Brien 2 gd? —¢2 
a ya | SN 
Mm? sin? be m? sin? 
and popse quently the Pe TiDEaa of ue fraction, or 


= 











L 
d? sin? x—-m®sin? L- _(ofsintc} sin) (xsinunm ¥ sin 1) 


—_> 
brea Fe 


¥2 


‘ m® cos? lm cos? (m* cosld-m®cosb) « (m®cosl—m®cos L. 
Whence, by CHInaCling fhe root, there results penaily 


d os sin bem? sin yr Got sin L—m sin t) 


“(micos lous cos a (m? cos 1—m3 cos L) 

This expression, which is simple and symmetrical, has been 
obtained without any developement into series, without any 
omission of terms on the supposition that they are indefinitely 
small, or any possible deviation from correctness, except what 
may arise from the want of coincidence of the circle of cur- 
vature at the middle points of the arcs measured, with the arcs 
themselves ; and this source. of error may be diminished.at 
pleasure, by diminishing the magnitude of the arcs measured : 
though it must be acknowledged that such a procedure may 
give rise to errors in the practice, which may more than coun- 
terbalance the small one to which we have just adverted. 

Cor. Knowing the number of degrees, or the parts of de- 
grees, in the measured arcs m, m, and their lengths, which 
are here regarded as the lengths of arcs to the circles which. 
have R, r, for radii, those radii evidently become. known in 
inaghitude. At the same time there are given the algebraic 
values of n andr; thus, taking r for example, and exterminat- 


: c?.d? 
ing e? and x2, there results rR = . There: 











(d2—e?) sin? 1)? 
, fore, by putting in this equation the known ratio of d toc, ° 
‘there will remain only one unknown quantity d or c, which ~ 
may of course be easily determined by the reduction of the- 
last equation ; and thus all the dimensions of the terrestrial, 
at will become known. 

aa General 


FIGURE OF THE EARTH. 43 
General Scholium and Remarks. | 


Beas dk ee 
i. The value aes 1, =o, is called the compression of the 
terrestrial spheroid, and it manifestly becomes known when 
ped, me 
the ratio zis determined. But the measurements of philoso- 


phers, however carefully conducted, furnish resulting com- 
pressions, in which the discrepancies are much greater than 
might be wished. General Roy has recorded several of these 
in the Phil. Trans. vol..77, and later measurers have deduced 
others. Thus, the degree measured at the equator by Bou- 
guer, compared with that of France measured by Mechain 
1 : 
339° also d=3271208 
teises, c==3261443 toises, d—c=9765 toises. General Roy’s 
sixth spheroid, from the degrees at the equator and in latitude 
: 1 | . . 
45°, gives 003" Mr. Dalby makes d = 3489932 fathoms, 
e=3473656. Col. Mudge d=3491420, e=3468007, or 7935 
and 7882 miles. The degree measured at Quito, compared 
with that measured in Lapland by Swanberg, gives compres- 


and Delambre, gives for the compression 


< 1 : : 
sion = cere Swanberg’s observations, compared with Bou- 
1 
uer’s give —_-—-.  Swanbere’s compared with the decree of 
Sia AOR AB Go Le ee : 


Delambre and Mechain 


307°4° 





Compared. with Major Lamb 


~. Aminimum of errors in Lapland, France, 


1 
ton’s degree ; 
Bree 307-17 





| : 1 : 
and Peru gives Soa-4' Laplace, from the lunar motions, finds 
compression an From the theory of gravity as applied 
; | oe oe | 
to the latest observation of Burg, Maskelyne, &c. ——~-. 
eo 309:05 
From the variation of the pendulum in different latitudes 


33570 Dr. Robinson, assuming the variation of gravity at 


ao 





_® This number 334-77 does not result from the variation of the pendulum in 
different latitudes, but is altogether erroneous in consequence of certain numeri- 
cal mistakes in La Place’s calculations. fe 


186 


94 dah aa sored 


1 iy 
| i 30" asker the compression 319" "Others give ae Very 


1 1 
ing fr OM Tae, gat to BiT bat far the greater number of otisere 


% iF 
vations differ but little from — SOP oN the computation from 


the phenomena of the precession of the equinoxes and the 
nutation of the earth’s axis, gives for the maximum list of” 
the compression. — 

2. From ihe various results of careful salcodlstinohedis it 
happens, as Gen. Roy has remarked, ‘‘ that philosophers are 
hot yet agreed in opinion with regard to the exact figure of, 
the earth ; some contending that it has no regular figure, that 
is, not such as would be generated by the revolution of a 
curve around its axis. Others have supposed it to be an 
ellipsoid ; regular, if. both polar sides should have the same 
‘degree of flatness ; but irregular if one should be flatter than 
the other. And lastly, some suppose it to be a spheroid dif- 
fering from the ellipseid, but yet such as would be formed by 
the revolution of a curve around its axis.’ According to the 
theory of gravity however, the earth must of necessity bave 
its axis approaching nearly to either the ratio of 1 to 680 or 
303 to 304; and as the former ratio obviously does not ob- 
tain, the figure of the earth must be such as to correspond 
nearly with the latter ratio. 

3. Besides the method above described, others have been 
proposed for determining the figure of. the earth by measure- 
ment. ‘hus that figure might be ascertained by the measure- 
ment of a degree in two parallels of latitude ; but not so ac- 
curately as by meridional arcs. Ist. Because, when the dis- 
tance of the two stations, in the same. parallel is measured, 
the celestial arc is not that of a parallel circle, but is nearly 
the arc ofa great circle, and always exceeds the arc that cor- 
responds truly with the terrestrial arc. 2dly, The interval 
of the meridian’s passing through the two stations must be 
determined by a time- -keeper, a very small error in the going 
of which will produce a very considerable error in the com- 
putation. Other methods which have been proposed, are, by 
comparing a degree of the meridian in any latitude, with a 
degree of the curve perpendicular. to the meridian in the same 
latitude ; ; by comparing the measures of degrees of the curves 
perpendicular to the meridian in different latitudes ; and by 


- eomparing an ar¢ of a meridian with an are of the parallel of 


latitude that crosses it. The theorems connected with these 
and some other methods are Hieon en by Professor Play- 
fair 





FIGURE OF "Tag F; ARTH, 85 
fair in the Edinburgh Tratisactions, vol. Veto whiew, todethor 
with the books meutioned at the end of the Istsection ut big 
chapter, the reader is: referred for much useful informatir 
on this bighly interesting subject, 

Having thus solved the chief problems connected. with 
Trigonometiical Surveying, the student is now presented with 
the following examples by way of exercise. 

‘Ex. 1. ‘She angle subtended by two distant objects at a 
third object is 66% 30 39"; one of those objects appeared un 
der an elevation of 26’ 47", the: other under a depression of 
1”, Required the reduced horizontal angle. Ans. 60° 30°37”, 

Ex: 2. Going along a straight and horizontal road which 
passed by a tower, 1 wished to find its height, and for this 
purpose measured two equal distances each of 84 feet, and at 
the extremities of those distances took three angles of eleva- 
tion of the top of the tower, viz. 36° 50’, 21° 24, and 14°. 
What is the height of the tower ? Ans. 53: 26-feet. 

Eity3. Investigate General Roy’s rule for the spherical ex~ 
eess, given in the scholium to prob. 8. | 

Ex. 4. The three sides of a triangle measured on the earth’s 
surface (and reduced to the level of the sea) are 17, 18, and 
10 miles ; what is the spherical excess ? 

Ex. 5. The base and perpendicular of another friangle are 
24 and 15 miles. Required the spherical excess. 

Ex, 6. Ina triangle two sides are 18-and, 23 miles, and they 

‘include an angle of 58° 24'36”. What is the spherical excess ! 

Ex. 7, The length of a base measured at an elevation of 38 

feet above the level of the sea is aA2R8 feet ; required the 
length when reduced to that level. 
_ Ex. 8. Given the latitude ofa eee A8° Bl'n the sun’s 
declination 18° 30'n, and the‘sun’s altitade at 10° 11™ 26s AM, 
52°35’; to find the angle that the vertical on which the sun is, 
makes with the meridian, | 

Ex, 9. When the sun’s longitude is 99° 13/43", what is his 
right ascension ? The obliquity of the elliptic being 23° 27’ 40”, 

fiz. 10. Required the longitude of the sun, when his right 

ascension and declination are 32° 4652” 1 and 13° 13’ 27”. x 
respectively. See the theorems in the echolium to prob. 12 

Ex, 11. The right ascension of the star « Ursa majoris is 
162° 50’ 34”, and the declination 62° 50'n : what are the longi- 
tude and latitude ? The obliquity of the ecliptic being as above. 

Ex. 12. Given the measure of a degree on the meridian im 

nN. lat, 49°3', 60833 fathoms, and of another in n. lat. 12932’, 
60494 fathoms ; to find the ratio of the earth’s axes. 

Ez. 13. Demonstrate that, if the earth’s figure be that of an 
oblate spheroid, a degree of the earth’s equator is the first of 

twe 


fe 


S 


96 _POLYGONOMETRY. 

two eceeonticuale between the last and first segnee) of 

latit 

ian Demonstrate that the degrees of the terrestrial meri- 
ian, in receding from the equator towards the poles, are in~ 

creased very nearly i in the duplicate’ ratio of the sine of the la- 


titade. | 
Ex. 15 If p be the measure of a degthe of a great circle 


‘perpendicular to a meridian ata certain point, m that of the 


corresponding degree on the meridian itself, and d the length 
of a degree on an oblique arc, that arc making an angle a 
Lh 


taeup) aint i Required 


with the meridian, then is d = 


a demonstration of this waren: 
ame 


PRINCIPLES OF POLYGONOMETRY,. 


Tue theorems and problems in Polygonometry bear an in- 
timate connexion and close analogy to those in plane trigono- 
metry ;-and are in great measure deducible from the same 
common principles. Each comprises three general cases. 

1. A triangle is determined by means of two sides and an 
angle ; or, which amounts to the same. by its sides except 
one, and its angles except two. In lke manner, any rectili- 
near polygon is determinable when all its sides except one, and 
all its angles except two, are known. 

2. A triangle is determined by one side and two angles ; 
that is, by its sides except two. and allits angles. So likewise, 
any rectilinear figure is determinable when all its sides except 


‘two, and all its angles, are known. 


3. A triangle is determinable by its three sides; that is 
when all its sides are known and all its angles, but three. In 
like manner, any rectilinear. figure is determinable by means 
of all its sides, and all its angles except three. 

In each of these. cases, the three unknown quantities may 
be determined by means ‘of three independent equations ; the 
manner of deducing which may be easily explained, after the. 
following theorems are duly understood. 


"THEOREM L 
In Any polygon, any One Side is Equal to the Sum of all 
‘The Rectangles of Each of the Other Sides drawn into the 
_ Cosine of the Angle made oe that Side and the Proposed 
 Side*. (os 
* ‘This theorem and: the following one, were announced by Mr. Lexel of Pe- 


tersburg, in Phil. Trans. vol. 65, p. 282: but they were fitet demonstrated by Dr. 
tie in Phil. Trans. vol 66, pa: 600. 
Let 


POLYGONOMETRY. 9% 


Let apcpger bea polygon: then 
will ar=aB. COS A-+BC . COS CB4. 
Fra -+.cp,. COS cD“ AF -+ DE. cos 
DE* AF--EF . COS EF* AF*, 

For, drawing lines from the se- 
veral angles, respectively paral- 
lel and perpendicular to ar; it 
will be 

Ab = AB. COS BAF, 

bc = BB = BC. COs CBB = BC . €08 CB, AF, 

cd = dp = cD. COS CD = CD. COS cD4 AF, 

de = «Ek = DE». COS DE¢ = DE. COS DE‘ AF, 

ef =....EF. COS EF€é = EF .COSEF* AF. 
But ar = bc + cd + de + er — ab; and ab, as expressed © 
above, is in effect subtractive, because the cosine of the ob- 
tuse angle BaF is negative. Consequently, 
ar=ac-+cd-+-de--er=ap . cos BAF-+BC . cos cB’ AF-+ &c, as 
in the proposition. A like demonstration will apply, mutatis 
mutandis, to any other polygon. 

Cor. When the sides of the polygon are reduced to three, 
this theorem becomes the same as the fundamental theorem 
in chap. ii, from which the whole doctrine of Plane Trigono- 
metry is made to flow. 





THEOREM II, 


The Perpendicular let fall from the Highest Point or Summit 
of a Polygon, upon the Opposite Side or Base, is Equal to 
the Sum of the Products of the Sides Comprised between 
that Summit and the Base, into the Sines of their Respec- 
tive Inclinations to that Base. . 
Thus, in the preceding figure, cc=csg . sin cB*Fa-+Ba . sin 

43 orcc=cp .sin cp‘ar-+pse. sin pE*aF+ rr. sing. This 

is evident from an Inspgction of the figure. 

Cor. 1. (n like manner pd=pe. sin DE‘AF-+EF , sin F, oF 
bd=cp . sin cB‘FA-+BA . SIN A—CD . Sin CD4aF. 

Cor. 2. Hence‘the sum of the products of each side, into 
the sine of the sum of the exterior angles, (or into the sine of 
the sum of the supplements of the interior angles), comprised 
between those sides and a determinate side, is = -++ perp. — 
perp. or =0. That is to say, in the preceding figure, 

AB. Sin A-+ BC. sin (A +8) -+- CD, Sin (a-+s+c)-+pE - sin 
(atete+p)+er.sn(aq-s+e+nd-+e) =0. 








* When a caret is put between two letters or pairs of letters denoting lines, 
the expression altogether denotes the angle which would be made by those two 
‘lines if they.were produced till they met, thus cg4#a denotes the jnclination of 
the line cg to Fa. 


Vou. NI, 14 Here 


98 _ POLYGONOMETRY- 


Here it j Se observed; that the sines of erates greater than 
180° af negative (ch. ii equa. vir). 
or. 3. Hence again, by putting for sin (44+ B). sin (a 4-8 


a! e), their values sin a . cos B + Sin B. cos, sin A. cos (B 
rd +c) + sin ero cos a, &c. (ch. ii equa. v), ae recollect- 


ing that tang = — = (ch. ii p. 55), we shall have, 


SIN A. istics cos B+cp . cos (8-+c)-+tDE . cos (B-+c-++p) 
+ &e.) + cos 4. (Bcsin B+ep) - sin (8-+c)--pE . cos (B-+c 
-+-p)-+&c.)=0; and thence finally. tan 180°—a, or tan BarF= 
Bo. sin B-+ cD. sin (B+4+-c) -FDE. sin(B=FC-++D) FEF. sin (p-+o# vp +8) 
AB + BC.COS B-+cD. cos (B +c) -+-bDE. cos (B-- C7 D)-+- EF. cos (B--C-++-D-+£) 
A similar expression will manifestly apply to any polygon.; 
and when the number of sides exceeds four, it is highly useful 
im practice. 

Cor. 4. Ina triangle anc, where the sides as, pc, and the 
angle ABC, OF ui PRE B, are known, we have 


. In B AB. SINB 
tan can == — >... 1) ten SS ee 


a .cOSB iy BC--AB . €OS B 
in both which expressions, the second term of the denomina- 
tor will become subtractive whenever the angle sxc is acute, 
or B obtuse. 


THEOREM IIE. 


The Square of Any Side of-a Polygon. is Equal to the Sum of 

the Squares of All the Other Sides, Minus Twice the Sum 

of the Products of all the other Sides Multiplied two and 
two, and by the Cosines of the Angles they Include. 


For the sake of brevity, let the sides e De C 
be represented by the small letters which 
stand against them in the annexed figure: é 
then, from theor. 1, we shall have the 
subjoined equations, viz. _ hi SESH 


a=b.cos ash +c. cosate + @. cos a*d, 
b—a.cos ah +c. cos b’ec + 2. cos 642, 

e =a.cos atc + 6. cos be + 2. cos cd. 
d—a.cosa*d+b.cos b40+¢.cosctd. | 
-Multiplying the first of these equations by a, the second by 6, 
the third by c, the fourth by 2; subtracting the three latter 
products from the first, and transposing 6?, c?, ¢?, there will 

Soha 
=? + ¢2?-+32 —2(be . cos b*c-+-b2 . cos b*d-+-cd . cos c*d). 
In ‘ike manner, 
eta? ates ey —2(ab. cos a4b+ad.cosa*d-+-be . cos b' }). 
&e. &ce. 
Or. 


Sas. 3 
iin 9g 


Or, since be=c, bd =c + p—1209, 69 =. we have 
a? = 6242+ 02 —2(be . cosc—be . cos(c+-n)+¢0~ cosp), 
¢? = a?-++ 62+ 0? —2(ab. cosp—b? . cos(a+s)+aé. coz), 
i ~ &e. &e. ) 
The same method applied to the pentagon azcpe, will give 


be. bd . + be. cos 
Past tp peta hE td. oe (os Dk be cs eka te} 


And a like process is obviously applicable to any number of 
sides ; whence the truth of the theorem is manifest. - 

_ Cor. The property of a plane triangle expressed in equa. 1 
ch. ii, is only a particular case of this general theorem. 


THEOREM IV. 


Twice the Surface of Any Polygon, is equal to the sum of 
the Rectangles of its Sides, except one, taken two and two, 
by the Sines of the Sums of the Exterior* Angles Contain- 
ed by those sides. © 
1. For a trapezium, or polygen of four 

sides. Let two of the sides as, pc, be 

preduced till they meet at rp. Then the 

trapezium azep is manifestly equal to the = \s 

difference between the triangles pap and 45>-—3e 

pee. But twice the surface of the trian- 
gle pap is (Mens. of Planes pr. 2 rule 2) ab. rp. sin P= 

{an+sp). (pc+cr) . sin Pp; and twice the surface of the tri- 

angle ppc is = BP . Pc . sin P: therefore their difference, or 

twice the area of the trapezium is = (az . pc-+aB . cP-} DC . 

BP). sin Pp. Now, in A esc, 








: ; EC .SINB 

sin P : sin B.: :.BC > PC, Whence Pe = ——_—___ 
sin P 

; M BC. Sin Cc 

sin Pp: SIN ¢ :: BC : PB, whence ps = —--——_. 
sin P 


Substituting these values of ps, pc, for them in the above 
equation, and observing that sin P = sin (PEC-+-PcE)=sin sum 
of exterior angles 8 and c, there results at length, | 
AE .EBC.SIN B 
spi surface = ( --aB . DC. sin (B-+c)- 
of apes: +8c.pc. sinc. 
Cor. Since ap . Bc . sin B = twice triangle asc, it follows 

that twice triangle acp is equal to the remaining two terms, viz. 
AB... Dc . sin (Bc) 


- twice area acD = 
. +Bc.Dc. sinc. 





* The exferior angles here meant, are those formed by producing the sides in 
tlie same manner as in th. 20 Geometry, and in cors. 1, 2, th. 2, of this chap. 


2. For 





e trapezium asco, and of the tri- B 
, Let the sides as, pc, as before, ? 
“- meet when produced at ep. Then, from f- \ 
the above, we have PC | a») 
wice area of l- AB. BC. SIN B 





the trapezium ap. pc. sin (B-+c) 
ABCD +s8c. pc. sinc. 
And, by the preceding corollary, 3 
Twice eee ; AP. DE. SID (e-+p) or sin (B-+c-++p) 
DAE +pP.DE. Sin D. 
AB. DE . sin (8-+c-+D) 
That istwice?  )-+-po. DE. Sin 
triangle pare §~ )-+BP. bE. sin (e+e+p) 
+cp . DE. Sin D. 
Bo. sine oy Le . SIN B 
sin sin (B +c) c)’ " sate +c)° 
BC. DE. sinc. ‘dntete-tn) , nome: .DE Sin BSIND 
sin (B-C) sin (B-+c) 
sing .sinn + sinc. sip (e+e+p) 
: and this ex- 
sin (B+c) nee 
pression by means of the formula for 4 arcs (art 30 ch. iii,) 
-becomes se. DE. sin (c-+p): Hence, collecting the terms, 
and arranging hea in the order of the sides, they become. 


AB. BC . SID B 
| sas «pe. sin (B+c) 


Now, sr = : therefore the last 





two terms become - 





= BC. 


Twice the area 
of the penta- 
gon ABCDE 


_ J +8 . ve . sin (B-+c+p) 

~ j+se. de. sine 

. Re . DE. sin (c-+p) | 
+-pc . DE. SiN D: 

Cor. Taking away from this expression, the Ist, 2d, and 
4th terms, which together make double the trapezium agcn, 
there willremain = 

Twice area of (AB. DE. Sin (p-+c-+p) 
the triangle } = +Be . DE. sin (c-++D) » 
DAE (+pc. . SIND. 

3, For a hexagon, as aBcpeEF. "The 
double area will be found, by suppos- 
ing it divided into the pentagon azcpE, 
and the triangle acr. For, by the last 
rule, and its corollary, we have, 





: 


POLYGONOMETRY. 104 


| AB 
| | 4 
Twice area of ae 
the pentagon ) = 

ABCDE , 


the triangle 
“AEF 


Twice area of : AP 


+-AB. 


.BC.SINB 

. CD. sin (B-++-c) 

DE. sin (B-+c-++p) 
.cD. sinc 

. DE. sin (c--p) 

. DE. sin D. 

. EF . sin (B-+c+p-+z) 
. EF . sin(p-FE) 

. EF . SIN E. 


. EF . sin (B-+c-++p-+-E) | 


the triangle } ={ ++pE. EF. sin£ 
AEF | +Be.er. sin (8-+c++D-++E) 
L-++cpe . EF. sin (p-++B). 
Now, writing for BP, cp, their respective values, 
Bo. sinc | Be. sin B 
sin (B-+tc) ane sin (B-Kc) ’ 
in the double areas of arr, will become 
sinc. sin (B-+-c-++p-++£)-+sin B. sin (D-FE) 
: sin (Bc) ; 
and this, by means of the: formula for 5 arcs (art. 30 ch. iii) 
becomes sc. EF sin(c+p-++z). Hence, collecting and pro- 
perly arranging the several terms as before, we shall obtain 


Or, twice area | | -+pce. BF. sin (p-+-5) 


the sum of the last two expressions, 


BC . EF 








( AB.BC.SiInB 
+aB. cp. sin (B-+c) 
-+aB . DE. sin (B-+c-+p) 
Twice the area Hee ai le (ste+pte) 
of the hexa- } = 2 qi ‘Shag tect . 
ta hnee } -FBc . DE. sin (cp) 
6° +c . eF. sin (c+D-+E) 
-+-cp. DE. sin D 
-hep . EF. sin (p--E) 
{ +pE. EF. sin E. 


4, Ia asimilar manner may the area of a heptagon be de- 
termined, by finding the sum of the areas of the hexagon and 
the adjacent triangle: and thence the area of the octagon, 
nonagon, or of any other polygon, may be inferred; the law 
of continuation being sufficiently obvious from what is done 
above, and the number of terms = Le ae 3 ? 


number of sides of the polygon is »: for the number of terms 
is evidently the same as the number of ways in which n— 1! 
quantities can be taken, two and two ; that is, (by the nature 
nm—=—1on—-2 . na 
2 





, when the 








- of Permutations) = 
Scholium. 


182 _ POLYGONOMETRY. 


’ Scholium., 
This curious theorem was first investigated by Sumon Pai 
lier, and published in 1789. Its principal advantage over the 
common method for finding the areas of irregular polygons 
~ as, that in this method there is no occasion to construct the 
figures, and of course the errors that may arise from such 
constructions are avoided. 
In the application of the theorem to practical purposes, the 
expressions above become simplified by dividing any propdsed 
polygon into two parts by a diagonal, and computing the sur- 
face of each part separately. 
Thus, by dividing the trapezium asco into two triangles, by 
the diagonal ac. we shall have , 
Twice area? _§ .aB.BC.SiINB 
trapezium | {= -+-cp. aD. sin D. 
The pentagon ascpe may be divided into the trapezium ABED ; 
and the triangle apz, whence 
AB. BC. SIN B 
Twice ons of? _ )-+az. vc. sin (8+c) 
pentagon ~~ )-FBc. pc. sinc 
-+--DE. AF. SINE 
Thus again, the hexagon may be divided into two trape- 
ziums, by a diagonal drawn from a to p, vale is to be the line 
excepted j in the theorem ; then will 
AB . BC. 61N B 
/ | tas . Dc. sin (B-++c) 
+Be.Dc. sinc 
+-DE. EF. sin E 
-+ne. aF. sin (e-+F) 
L-Fer. aF. sin F. 
And Tasty, the heptagon may be divid- 
ed into a pentagon and a trapezium, the 
diagonal, as before, being the excepted p 
line : so will the double area be express-  ~ 
ed by 9 ahaa of 15 products, thus : c 


Twice area of 
hexagon me 






{ aB.BC. SIN B 
+4B. cp. sin (B-+c). 
a? . DE. sin (B-++c-+p) 
Twice area He BC. cD. Sin c 
-+-Be. DE. sin (c4+-p 
af Peprein -++-cp. DE. Sin « ) 
-+-EF.FG.SINF 
+eFr.ca. sin (F+c) 
Piers pace 
The same method may obviously be extended te other poly- 
gons, with great ease and simplicity. 
, tt 





Hy enc 





POLYGONOMETRY. 103 


it often happens, however, that only one-side of a polygon 
can be measured, and the distant angles be determined by in- 
tersection ; in this case the area may be found, independent 
ef construction, by the following problem. 


PROBLEM f, 


Given the Length of One of the Sides of a Poly gok: and the 

- Angles made at its two extremities by that Side and Lines 
drawn to all the Other Angles of the Polygon: to find an 
_ Expression for the Surface of that Polygon. 


Here we suppose known re ; ali 
ara=a', BPa=b’, cpa = ¢!, DPQ = d' ; 
AQP=a, nape"; cep=c’, par= a”. 
Now, ain PAQ = sin (a’-+-a") ; sin PEG 


= sin (b-+6"), 





Therefore, sin (@ +a”): PQ: : sin a”; pa = -——- 


° ’ WwW. - : Si Mt : § vides oe 
And, ...sin (6+ 6") >: sin b giceste sin (0-0) ° 


But, triangle arp=ar .. Pp. 4 sin APB —= 2 AP, PB, sin (a oy), 


u F ae ee sin a a’. sin b”. sin (a! — bf bt) 

~sin ach . sin ot . sin w sin e 
sin (6’-+ 6”). sin (c’ + ¢”)’ 
sinc”. sin d’. sin sin (c’ — d’) 
sin (c+ ¢” +e”). sin’ adi whe nee 
&c. &c. &c. 





In like manner, A Brco=-1 PQ? . 


= 2 
A cpp=} PQ? . 





sin da’ 


= @P.PpD.1 sin pPea=P - PQ .sin d’ 
hore 2 SOc Sin (dp py ie 


sin d’. sin da’ 


==l 2. . 
gpk ete AE Ogee iain. (dE +d’) Consequently, 
(sin a” . sin 6”. sin n (a —V) 
sin (a’ + a!) . si sin (6’ n (b’+-b") 
; sin b”. sin c’ . sin n (b= —¢'} 
j 7 M1 Tw, 
Surface papcpg = 4 Pa®. $n gin (8 a wh ++ 6") . sin 1 (a - +e") 


sinc” ..sin d”.. sin (¢ —d’) —d/) 
sin (¢ tr ce’). sin (d’+d") 
sin a ~sind’ | 
igcepoerr 


sin (d+d%) +d”) 7 
€ 


yh, 


104 POLYGONOMETRY.- 


The same method manifestly applies to polygons of any 
number of sides: and all the terms except the last are so per- 
fectly symmetrical, while that last term is of so obvious a form, 
that there cannot be the least difficulty in extending the for- 


mula te any polygon whatever. 
-. PROBLEM I. 


Given, in a Polygon, All the Sides and Angles, except three ; 
to find the unknown Parts.. — 

This problem may be divided into three porerilicasba, as 
shown at the beginning of this chapter: but the analytical so- 
lution of all of them depends on the same principles ; and 
these are analogous to those pursued in the analytical investi- 
gations of plane trigonometry. In polygonometry, as well as 
trigonometry, when three unknown quantities are to be found, 
it must be by means of three independent equations, involv- 
ing the known and unknown parts. These equations may be 
deduced from either theorem 1, or 3, as may be most suited 
to the case in hand ; and then the unknown parts may each be 
found by the usual rules of extermination. 


For an example, let it be supposed D 
that in an irregular hexagon aBcver, E 
there are given all the sides except C 
AB, BC, and all the angles except 8; to Er 
determine those three quantities, 
Pe B 


The angle B is ad: equal to (2n — 4) right angles — 
(a-+ ¢ is p+e-+F); n being the number of sides, and the 
angles being here supposed the interior ones. 

Let an =z, sc =y: then by th. 1, 
r= Yy.CcOS B+ DC. Cos aB‘cD + DE - COS AB“ED, 

-+- EF. cos AB “EF -- AF . COS AB44P ; 
y= 2. COS B+ AF. cos BC AF + FE, COS BC“FE, 
+ DE.cos BCADE + pc . COs BC*cD. 

In the first of the above equations, let the sum of all the 
terms after y . cos B, be denoted by c; and in the second the 
sum of all those which fall after 2 .cos 8, by d; both sums 
being manifestly constituted. of known terms: and let the 
known co-efhicients of x and y be mand a respectively. Then 
will the preceding equations become 

L=ny + C....y=me + d, 
Substituting for y, in the first of the two latter equations, its 
value in the second, we obtain 2 =mnx + nd-++c. Whence 
there will readily be found 
nd+te F me +d 


‘eee ine 1—snn 





Thus 


POLYGONOMETRY. 105 


Thus as and se are determined. ‘Like expressions will serve 
for the determination of any other two sides, whether. conti- 
guous or not: the co-efficients of x and y being designated by 
different letters for that express purpose ; which would have 
been otherwise unnecessary in the solution of the individual 
case proposed. : 

Remark. Though the algebraic investigations commonly 
lead to results which are apparently simple, yet they are often, 
especially in polygons of many sides, inferior in practice to 
the methods suggested by subdividing the figures. The fol- 
lowing examples are added for the purpose of explaining those 
methods: the operations however are merely indicated ; the 
detail being omitted to save room. 


EXAMPLES, — 


Ex. 1. Ina hexagon ascper, all the sides except AF, and 
all the angles except a and Fr, are known. Required the un- 
known parts. Suppose we have 


ap== 1284 — Ext. ang. _. Whence 

Bc == 1782 B == 329 B-+-+c = 80° x) 
cp == 2400 c = 48° B-+c-+pD == 132° 

DE = 2700 p== 52° — B-Fo-+-p-+E = 198° 

EF = 2860 E == 66° AF = 162°. 


Then, by cor. 3 th. 2, tan Bar = 

BC. Sin B -- cp. sin (B--c)4-pz. sin (B+-c-+4-D) + er. sin (Bc D+) 
AB-} BC . Cos B+ cD. Cos (B+-c) ++ DE. cos (B+ C + D) +f EF . Cos (B-PC-++-D- EF) 
Bc . sin 32° -+- cp. sin 80° + be. sin 132°-+ EF. sin 198° 
AB BC . COS 32°-+-cp . cos 80°-+-pE . cos 132°-++ EF . cos198¢ 

Bc . sin 32°--cp . sin 80°-++-DE . sin 48% — EF . sin 182 

~~ AB-+Bc . cos 32°-+ cp . cos 80° — DE. cos 48° —EF.cos18° 
Whence sar is found 106° 31’ 38”; and the other angle arr 
== 91° 28' 22”. So that the exterior angles a and ¥ are 73° 
28' 22’, and 88° 31’ 38” respectively: all the exterior angles 
making 4 right angles, as they ought todo. Then, all the 

angles being known, the side aF is found by th. 1=4621°5. 
If one of the angles had been a re-entering one, it would. 
have made no other difference in the computation than what. 
would arise from its being considered as subtractive. 
Ex. 2. Ina hexagon ascper, all the sides except ar, and 
all the angles except c and B, are known: viz. 





aBp==2400 Ex. Ang. We shall have, by th. 2 cor. 1, 
Bo=2700 a= 54° oy Ah ne 
Cp=s400> Ye =2G2% 08 2S } DE.sin(E--F 
DE=3500.  e=64° Fae: ie les oF. } -+-erF. sin F, 
gr==3750 ° p= 720° -+-cp > SIN a-+-B-+-c) 


Vor. I. 15  There- 


106 - POLYGONOMETRY. 


aby ? oO —aB v sin’ 54° 
r tiles : tok AL Y—Be. sin- 116° 
Therefore, cP . sin (116°+-c) = Sil es win 136°" 
HRS pus eae +EF.sin 72°. 
__ § . 149° 23! 26", 
Or, 116°+c= ) 1 330 36’ 34". 
The second of these will give for c, a re-entering angle ; 
the first will give exterior angle c==33° 23’ 26’, and then will 
p=14° 36 34. Lastly, et 
—AB. cos 54° 


-++Bc .cos 64° . | 
ar= ¢ +cep. cos 3093634" > =3885:905. 
--pE . cos 44° 4" 


HEF. COS 72° 
Ex. 3. In a hexagon ascper, are known, all the sides ex-- 
cept aF, and all the angles except 5 and £ ; to find the rest. 
Given an = 1200 Exterior angles a = 64° 


Bc = 1500 

cp = 1600 . C= 728 
DE = 1800 D = 75% 
EF = 2000 


F = 84°. 

Suppose the diagonal se drawn, dividing the figure into two 
trapeziams. Then, in the trapezium scns the sides, except 
BE, and the angles except Bp and ze, will be known; and these 
may be determined as in exam. 1. Again, in a trapezium 
ABEF, there will be known the sides except, ar, and the angles 
except the adjacent ones p and E. Hence, first for BcpE: 
(cor. 3 th. 2), . 

cD. sin c-+pE. sin (c--p)  __ 

BC--CD . COS C--DE . COS (c-+p) 

cD . sin 72°--pg . sin 147° -cp . sin 72°-F-nE . sin 33° 

BC-+-cD.cos 72°-+-DE.cos 147° Bc-+ep - COS 72°-—DE . cos 33° 

Whence cre = 799% 2’ 1”; and therefore peB = 67° 57 59”. 
% Bc . cos 79° 9" 1” 

Then rz = ss ..cos 7° 2’ o == 2548:°581. 


tan cBE= 


-+-pe. cos 67°57'59” 

Secondly, in the trapezium ABEr, . 
AB. SIN A-PBE. Sin (A+B) == EF. sin F: whence’ 

we -\ _EF-SINF—AB.SINB . ¢ 90° 55/54, 

PEACE) Se ee } 159° 4° 6”, 

Taking the lower of these to avoid re-entering angles, we 
have B (exterior ang.) = 95° 4’6”: ape = 84° 55/54”; ren 
63% 4'6”: therefore apc = 163° 57’ 55”; and rep=131°2" ; 

and 





ROLY SONDME TRY. 107 


and consequently the exterior ane at B and £ are 16°2 5’ 
and 48° 57’ 55” respectively.. 

Lastly, AF =—AB . COS AmBE . cos s (a-te)— EF COS F == — 
ap. cos 64° -- BE. cos 20° 55'°54”— er . cos 84°= 1645-292. 

Note. The preceding three examples comprehend all the - 
varieties which can occur in Polygonometry, when all the sides 
except one, and all the angles but two, are known. ‘The un- 
known angles may be about the unknown. side ; or they may 
be adjacent to each other, though distant from ‘the unknown 
side; and they may be remote from each other, as well as 
from the unknown: side. 

Ex. 4. {na hexagon aBcper, are known all the angles, and 
all the sides except ar and en : to find those sides. 


Given aB = 2200 Ext. Ang. A= 96° 


Be = 2400 _ B= 54° 
: “»~ C= 20° 
pE = 4800 p= 24° 
EF = 5200 E= 189 
F:==148°? 


Here, reasoning from the principle of cor. th. 2, we have, 


AB.sin 96° ‘or AB.sin 84° 
. DE.sin166° ss DE. sin14° 
Bc.sinl50 ‘ face einl43° Bc.sin 30 4ar. " sin329 


+-cp.sin170° ++ cp.sin 10° 
Whence ¢ px.sinl4°.cosec10°—az.sin84%.cosec 10° = 304558 
Roce con sin32°.cosec10°—zc.sin30°. “cosec 03 oe : 

And ; DE.sin24°.cosec 10°—cB.sin20° 14874-99. 
AF = ) 4ur.sin42°.cosecl0°—za.sin74° § ; 

Ez. 5. In the nonagon apcpereut, all the sides are known 

and all the angles except a, p, c : it is required to find those 
angles. 


Given ap == 2400 re = 3800 Ext, ang. B == 40° 


Bc = 2700 co = 4000 C S22 
cp = 2800 ar = 4200 E = 369? 
pE = 3200 1 = 4500 F = 45° 
EF = 3500 H = 489 


1 == 609, 


Suppose diagonals drawn to join the 
unknown angles, and dividing the poly- 
gon into three trapeziums and a triangle ; 
as inthe marginal figure. Then, 

Ist. In the trapezium azcp, where ap, 
and the angles about it are unknown we 
have (cor. 3. th. 2). 





108 . POLYGONOMETRY. 


_ Bc. sinB-+cp. sin(peec) Bo, sin » sin 408 te oD 6D sin 70° 
ee ABP c.cos BF cp. cos meee AB + BC . cos 40° 40° cp cD .cos 72° 
Whence: BAD = 39° 30! 42", cpa == 32° 29' 187’. 

AB. cos 39° 30 42) 
And ap=<¢-+sc.cos 0 2918 5 = 6913°292. 
-+-cp.cos 32. 29 18 
_ Qdly. In the quadrilateral pere, where ve and the angles 
about it are unknown; we have © 
EF. it ke sin (E+ ¥F) eRe. sin abe: Fc . sin 81° 


tan Epe= DE + EF .COSE++-FG.COS(E4+F) DE-LEF. cos 36°-+- FG . cos 818 
Whence evc=41° 14’ 53”, rap==39° 45 7”. 
DE. cos 41° 14’ 53” ‘3 
And po= (¢ +er.cos &° 14’ 53” 5 =8812°-803. 
-+re.cos 39° 45’ 7” 

3dly. Inthe trapezium GHIA, an exactly’similar process gives 
HGA == 50° 46’ 3”, rac =47° 13’ 7”, and ac = 9780-591. 

4thly. In the triangle ane, the three sides are now known, 
to find the angles ; viz. pac=60° 53’ 26”, agp = 43° 15’ 54”, 

abDG = 75° 50'40'’. Hence there results, lastly, 

TAB ==47° 13’ 7”’4-60° 53’ 26”4-39° 30°42" = 147° 37) 15’, 

CDE=32° 29° ,18"-+-70 50’ 40’-+41° 14/53” = 149° 34’ 51", 

FoH=39° 45 '7-4+-43° 15! 54°-4+50° 4653” = 133° 47 54’. 

Consequently, the required exterior angles are A=32°22'45", 

d= 30° 25' 9”, gc = 46° 12' 6". 

Ex. 6. Required the area of the hexagon in ex. 1. 

Ans. 16530191. 

Ex. 7. Ina quadrilateral ABCD, are given aB=24, sc=30, 
cp= 34; angle anc = 92° 18’, pcp==97° 28’. Required the 
side ap, and the area. 

Ex. 8, In prob. 1, suppose Pe=—2538 links, ah the angles 
as below ; what is the area of the field ancpar ? — 
APQ==89° 14’, ppa==68° 11’, cpa=36° 24’, ppga= 19°57’; 
AQP==25° 18’, BaP=69° 24’, cap==94° 6, nars=121°18'. 

















OF, 


{109 7 


OF MOTION, FORCES, &e. 
“DEFINITIONS. 


Art. 1, BODY is the mass, or quantity of matter, in any 
material substance ; and it is always proportional to its weight 
or gravity, whatever its figure may be. 


2. Body is either Hard, Soft, or Elastic. A Hard Body 
is that whose parts do not yield to any stroke oy percussion, 
but retains its figure unaltered. A Soft Body is that whose 
parts yield to any stroke or impression, without restoring 
themselves again ; the figure of the body remaining altered. 
And an Elastic Body i is that whose parts yield to any stroke, 
but which presently restore themselves again, and the, body 
Eeeains the same figure as before the stroke. 


We know of no bodies that are absolutely, or perfectly, 
either hard, soft, or elastic ; but all partaking these proper- 
ties, more or less, in some intermediate degree. 


3. Bodies are also either Solid or Fluid... A Solid Body, 
is that whose parts are not easily moved among one another, 
and which retains any figure given.to it. But a Fluid Body 
is that whose parts yield to the slightest impression, being 
easily moved among one another ; and its surface, when left 
to itself, is always observed to settle i in a smooth plane at the 
top. 


4. Density is the proportional weight or quantity of mat- 
ter in any body. So, in two spheres, or cubes, &c. of equal 
size or magnitude; if the one weigh only one pound, but the 
other two pounds ; then the density of the latter is double the 
density of the former; if it weigh 3 pounds, its density ts 
triple ; ; and so on. 


5. Motion is a continual and successive ‘Dhante of place.—— 
If the body move equally, or pass over equal spaces in equal 
times, it is called Equable or Uniform Motion. But if it in- 
crease or decrease, itis Variable Motion ; and itis called Ac- 
_ celerated Motion in the former case, and Retarded Motion in 
the latter—Also, when the moving body is considered with 

respect 


110 OF MOTION, ‘FORCES, &c, 


respect to some other body at rest, it is said to be Absolute 
Motion. But when compared. with others in motion, it is 
icq Relative Motion. 


6. Velocity, or Celerity, is an affection of motion, by 
Which a body passes over a certain space in a certain time. 
Thus, if a body in motion pass uniformly over 40 feet in 4 
seconds of time, it is said to move with the velocity of 10 feet 
per second ; and so on. 


7. Momentum, or Quantity of Motion, is the power or 
force in moving bodies, by which they continually tend from 
their present places, or with which they strike any obstacle 
that opposes their motion. - 


8. Force is a power exerted on a body to move it, or to 
stop it. If the force act constantly, or incessantly, it is a 
Permanent Force: like pressure or the force of gravity. But 
ifit act instantaneously, or but for an imperceptibly small time, 
itis called Impulse, or Percussion : like the smart blow of a 
hammer. é 


9... Forces are also distinguished into Motive, and Accele- 
rative or Retarding. A Motive or Moving Force, is the 
power of an agent to produce motion ; and it is equal or pro- 
portional to the momentum it will generate in any body, when 
acting, either by percussion, or r for a certain time asa perma- 
nent force. _ 


10. Accelerative, or Retardive Force, is commonly. un- 
derstood to be that which effects the velocity only ; or it is 
that by which the velocity is accelerated or retarded ; and if 
is equal or proportional to the motive force directly, and to 
the mass or body moved inversely. —So, if a body of 2 pounds 
weight, be acted on by a motive force of 40; then the accele- 
rating force is 20. But if the same force of 40 act on an- 
other body of 4 pounds weight; then the accelerating force 
in this latter case is only 10; and: so is but half the fontaers 
and will produce only half the velocity. 


“11. Gravity, or Weight, is that force by which a Ma 
- endeavours te fall downwards. It is called Absolute Gravity, 

when the body is in empty uae and Relative Gravity, 
when emersed ina fluid. : 


jz: Specific Gravity is the ppepbntio of the weights of 
different bodies of equal magnitude ; ane SO is proportional 


to the ey of the body. 
AXIOMS. 


GENERAL LAWS OF MOTION. 111 


_ AXIOMS. 

13. Every body naturally endeavours to continue in its 
present state, wncver it be at rest, or moving uniformly in a 
right line. 

14. The Change or Alteration of Motion, by any external 
force, is always proportional to that force, and in the direc- 
tion of the right line in which the force acts. 

15. Action and Re-action, between any two bodies, are 
equal and contrary. -That is, by Action and Re-action, equal 
changes of motion are produced in bodies acting on each other ; 


and these changes are directed towards opposite or contrary 
parts. 


GENERAL LAWS OF MOTION, Ke. 


PROPOSITION I. 


16. The Guantiy of Matter, in all bodies, is in the Compound 
Ratio of their Magnitudes and Densities. ? 


Tuar is, b is as md; where b denotes the body or Lega 
ty of matter, m its magnitude, and d its density. 

For, by art. 4, in bodies of equal magnitude, the mass or 
quantity of matter, is as the density. But, the densities re- 
maining, the mass is as the magnitude : that is, a double: mag- 
nitude contains a double quantity of matter, a triple magnitude, 
atriple quantity, and so on. Therefore the mass is in the 
compound ratio of the magnitude and density. | 
17. Corol. 1. In similar bodies, the masses are as the den- 
sities and cubes of the diameters..or of any like linear dimen- 
sions.—F or the magnitudes of bodies are asthe cubes of the 
diameters, &c. 

18. Corol.. 2: The masses are as the BS and speci- 
fic gravities.—For, by art. 4 and 12, the densinies of bodies 
are as the specific gravities. 

19. Scholium. Hence, if b denote any body, or the quan- 


. tity of matter in it, m its magnitude, d its density, g its specific 
gravity, 


Vy 


412 ‘OF MOTION, FORCES, &c. 


gravity, and a its diameter or other dimension ; then, « (pro- 
nounced or named as) being the mark for general proportion, 
from this proposition and its corollaries we have these gene- 
ral proportions : 


b o md o mg oe asd, 


bin be Ma 
Ce eRT in) ore 


PROPOSITION II. 


20. The Momentum, or Quantity of Motion, generated by a 
Single Impulse, or any Momentary Force, ts as the Generating 
Force. 


Luar is, m is as f ; where m denotes the momentum, and | 
F the force. 

For every effect is proportional to its. adequate cause. So 
that a double force will impress a double quantity of motion ; 
a triple force, a triple motion; and soon. That is, the mo- 
tion impressed, iz as the motive force which produces it. 


PROPOSITION III. 


. The Momenta, or Quantities of Motion, in moving Bodies, 
are in the Compound Ratio Yy the Masses and Velocities. 


That is, m is as dv. 


For, the motion of any body being made up of the mo- 
tions of all its parts, if the velocities be equal, the momenta 
will be as the masses ; fora double mass will strike with a 
double force; a triple mass, with a triple force, and so on. 
Again, when the mass is the same, it will require a double 
force to move it with a double velocity, a triple force with a 
triple velocity, and so on; that is, the motive force is as the 
velocity ; but the momentum impressed, is as the force which 
produces it, by prop. 2; and therefore the momentum is as 
the velocity when the mass is the same. But the momentum 


was found to be as the mass when the yelocity is the same, 
7 Consequently 


». 


GENERAL LAWS OF MOTION, 113 


Consequently, when neither are the same, the momentum is 
in the compound ratio of beth the mass and velocity. 


PROPOSITION IV. 


22. In Umform Motions, the Spaces described are in the Come 
pound Ratio of the Velocities and the Times of their Descrip- 
won. 


That is, s is as tv. 


For, by the nature of uniform motion, the greater the ve- 
locity, the greater is the space described in any one and the 
same time ; that is, the space is as the velocity, when the 
times are equal. And when the velocity is the same, the space 
will be as the time ; thatis, in a double time a double space 
will be described ; ina triple time, a ttiple space ; and so on. 
Therefore universally, the space is in the compound ratio of 
the velocity and the time of description. 

23. Corol. 1. In uniform motions, the time is as the space 
directly, and velocity reciprocally ; or as the space divided 
by the velocity. And when the velocity is the same, the time 
is as the space. But when the space is the same, the time 
is reciprocally as the velocity. . 

24. Corol. 2. The velocity is as the space directly and the 
time reciprocally ; or.as the space divided by the time. And 
when the time is the same, the velocity is as the space. But 
when the space is the same, the velocity is reciprocally as 
the time. | 


Scholium. 


25. In uniform motions generated by momentary impulse, 
let b = any body or quantity of matter to be moved, 
f= force of impulse acting on the body 6, 
v == the uniform velocity generated in 64, 
m == the momentum generated in }, | 
$ == the space described by the body 4, 
t == the time of describing the space s with the veloc. v. 


Then from the last three propositions and corollaries, we 
have these three general proportions, namely f «x m,m & 
bv, and s « tw; from which is derived the following table of 
. the general relations of those six quantities, in woiform mo- 
tions and impulsive or percussive, forces : 

You. I}. 16 fe a 


114 OF MOTION, FORCES, &c. 
fx«emabreae a8 


: bs 
mafia bv ae —. 


m t t 
Leet ig ™, 


ta 


By means of which, may be resolved all questions relating 
e uniform motions, and the effects of momentary or vatiaine 
orces: 


PROPOSITION Vv. 


26. The Momentum generated by a Constant and Uniform Force 
acting for any Time, is in the ree Ratio of the Force 
and Time of Acting. : 


That ie m is as ft. 


For, supposing the time divided into very small parts, by — 
prop. 2, the momentum in each particle of time is the same, 
and therefore the whole momentum will be as the whole time, 
or sum of all the small parts. But by the same prop. the mo- 
mentum for each small time is also as the motive force. Con- 
sequently the whole momentum generated, is in the compound 
ratio of the force and time of acting. 

27.. Corol. 1.. The motion, or momentum, lost or destroyed 
in any time, is also in the compound ratio of the force and © 
time. For whatever momentum any force generates ina given 
time ; the same momentum will an equal force destroy in the 
same or equal time ; acting in a contrary direction. 

And the same is true of the increase or decrease of motion, 
by forces that conspire with, or oppose the motion of bodies. 

28. Corol. 2. The velocity generated, or destroyed, in any 
time, is directly as the force and time, and reciprocally as the 
body or mass of matter.—For, by this and the 3d prop. the 
compound ratio of the body and velocity, is as that of the force 
and time ; and therefore the velocity is as the force and time 
divided by the body. Andif the body and force be given, or 


constant, the velocity will be as the time. 
PROPOSITION 


GENERAL LAWS OF MOTION. 115 


\ 


PROPOSITION VL 


29, The Spaces passed over by Bodies, urged by any Constant 
and Uniform Forces, acting during any Times, are in the com- 
pound Ratio of the Forces and Squares of the Times directly, 
and the Body or Mass reciprocally. 

Or, the Spaces are as the Squares o the Times, when the Force 
and Body are given. 


e e {2 : se 
Tuar is, sis ale, or as #2 when f and b are given. For, 


let v denote the velocity acquired at the end of any time ¢, 
by any given body 6, when it has passed over the space s. 
Then, because the velocity is as the time, by the last corol. 
therefore 1 v is the velocity at 14%, or atthe middie point of 
the time; and as the increase of velocity is uniform, the 
same space s will be described in the same time ¢, by the 
velocity 4 v, uniformly continued from beginning to end. 

But, in uniform motions, the space is in the compound ratio 
of the time and velocity ; therefore s is as } tv, or indeed s== 


f : t 
itv. But, by the Jast corol. the velocity v is as be or as 


the force and time directly, and as the body reciprocally. 
i f3 

Therefore, s, or 4 tv, is as; that is, the space is as the force 
and square of the time directly, and as the body reciprocally. 
Or s is as t?, the square of the time only, when 6 and f are 
given. 

30. Corol. 1. The space sis also as tv, or in the esriacnell 
_ ratio of the time and velocity; & and fbeing given. For, 
s==j}tv is the space actually described. But tv is the space 


* which might be described in the same time ¢, with the last ve- 


locity v, if it were uniformly continued for the same or an 
equal time. Therefore the space s, or Ltv, which is actually 
described, is just half the space tv, which would be described, 
with the last or greatest velocity, uniformly continued for an 
equal time ¢. | 

31. Corol. 2. The space s is also as v?, the square of the 
velocity ; because the velocity v is as the time ¢. 


Scholium. 


32. Propositions 3, 4, 5, 6, give theorems for resolving all 
questions relating to motions uniformly accelerated. Thus, 


put 


116 OF MOTION, FORCES, & 


6 = any body or quantity of matter, 
f = the force constantly acting on it, 
 ¢ = the time of its acting, 
_» =the velocity generated in the time ¢, 
_ == the space described in that time. 
m == the momentum at the end of the time. 
Then, from these fundamental relations, m « bv,m « ft; 


¢ « tv, and v - , we obtain the follewing table of the ge- 


neral relations of uniformly accelerated motions : 


bs s > ft2v ale 
mabe fie an alry VJ bfs « /bfiv: 
t f¢2 243 2 2 
NUE ER LOA mn hag CONN ine ot 
D  § s $ ms 


—m bv mo ms m2 os m2 bv? : bs 
OC Oe Oe a te 
f t8h¢ $ t2v bs bte $ {2 
Ss t m ms sm? s 2st 
Dee at ne ——— © Et. 
BO BOR San oD SO ean? 
fe: mt feo)! mo. mt. bp?) mao. 


ith Paton he: b a he Yaad ee 7 
pte, We ee as pee ms mn he 
Df Rife ante Pe MFO BPD 


33. And from these proportions those quantities are to be 
left out which are given, or which are proportional to each 
other. Thus, if the body or quantity of matter be always 
the same, then the space described is as the force and square 
of the time. And if the body be proportional to the force, 
as all bodies are in respect to their gravity ; then the space 
described is as the square of the time, or square of the velu- 


city ; and in this case, if F be put ae the accelerating 


force ; then will 


v 
s « tv c Ft? « —, 
F 


$ 
Aaa Dale ie 2 


& 
¢ 


S 
a a ae 


THE 


GENERAL LAWS OF MOTION. int 


THE COMPOSITION AND RESOLUTION OF 
FORCES. 


34. Composition or Forces, is the uniting of two or more 
forces into one, which shall have the same effect ; or the find- 
ing of one force that shall be equal to several others taken 
together, in any different directions. And the resolution of 
Forces, is the finding of two or more forces which, acting in 
any different directions, shall have the same effect at any given 
single force. 


PROPOSITION VII. 


35. If a Body at a be urged in the Directions as and ac, by 
any two Similar Forces, such that they would separately cause 
the Body to pass over the Spaces ap, ac, in an Equat Time ; 
then if both Forces act together, they will cause the Body to 
move in the same Time, through av the Diagonal of the Pa- 
rallelogram aBcp. 


Draw cd parallel to as, and bd aral- 
Jel to ac. And while the body is carried 
over ab, or cd by the force in that direc- 
tion, Jet it be carried over bd by the force 
in that direction ; by which means it will 
be found at d. Now, if the forces be 
impulsive or momentary, the motions will D 
be uniform, and the spaces described will be as the times of 
description ; 





theref. ab orcd ; ap orcp:: time in ab ;: time in ap, 
and 6d or ac : BD or ac: : time in ac : time in Ac; 
but the time in ab: = time in ac, and the time in ap =: | 


time in ac; therefore ab : bd: : ap: Bp by equality ; hence 
the point d is in the diagonal ap. 

And as this is always the case in every point d, d, &c. there- 
fore the path of the body is the aetiah line “ad or the di- 
agonal of the parallelogram. 

But if the similar forces, by means of which the body is 
moved in the directions ap, ac, be uniformly accelerating 
ones, then the spaces will be as the squares of the times; in 
which case, call the time in bd or cd, t, and the time in az or 


ac, T : then | 
5h will be ab orcd: apoorcp:: t? : T2 
and - bd or ac: Bp or ac:: t? : 


theref, by equality, ab: bd: : ap: BD; 
and so the body i is always found in the diagonal, as before. 
36. Corol, 


Py . 


Pe 


# 


ot OF MOTION, FORCES, ke. 


2 | 36. Corol: 1. If the forces be not similar, by which the 
*’, body is urged in the directions ax, ac, it will move in some 
curved line, depending on the nature of the forces. 


37. Corol. 2..Hence it appears that the body moves over 
the diagonal ap, by the compound motion, in the very same 
time that it would move over the side as, by the single force 
impressed in that direction, or that it would move over the 


- side ac by the force impressed in that direction. . 


38. Corol. 3... The forces in the directions,azB, ac, AD, aré 
respectively proportional to the lines an, ac, aD, and i in these 
directions. 


39. Corol. 4. The two oblique forces 
AB, ac, are equivalent to the single di- 
rect force ap, which may be compound- 
ed of these two, by drawing the diagonal 
of the parallelogram. Or they are equi- 
valent to the double of ar, drawn to the 
middle of the line nc. And thus any 
force may be compounded of two or more 
other forces ; which is Epes catiing of the expression compe- 
sition of forces. 





40. Exam. Suppose it were 
required to compound the three ~ 
forces ap, ac, ap; or to find the 
direction and quantity of one 
single force which shall be equi- 
valent to, and have the same ef- 
fect, as if a body a were acted ; 
on by three forces in the directions ab, ac, ap, and pro- 
portional to these three lines. First reduce the two ac, ap, 
to one ax, by completing the parallelogram apgc. Then re- 
duce the two ar, 4B to one ar by the parallelogram arrs. 
So shall the single force ar be the direction, and as the quan- 
tity, which shall of itself produce the same effect, as if all the 





. three a8, ac, ap acted together. 


41: Corol. 5. Hence also any single 
direct force ap, may be resolved into, 
two oblique forces, whose quantities 
and directions are an, ac, having the 
game effect, by describing any paralle- - 
logram whose diagonal may be ap: and 
this is called the resolution of forces, 
So the force ap may be’ resolved into 
the two ap, ac by the parallelogram 





ABDC $ 


GENERAL LAWS OF MOTION. = wig 


aspc ; or into the two ar, Ar, by the parallelogram AEDF 5 
and so on, for any other two. And each of these ma 
solved again into as many others as we please. 





42, Corol. 6. Hence too may be 
found the effect of any given force, 
in any other direction, besides that 
of the line in which it acts; as of the 
force az in any other given direction 
es. For draw ap perpendicular to 
cp; then shall ps be the effect of the 





force az in the direction cs. For the given force as 
valent to the two ap, ps, or az; of which the former ap, or 
mB, being perpendicular, does not alter the velocity in. the di- 
rection ce; and therefore pz is the whole effect of Ax in the 
direction cz. Thatis,a direct force expressed by tht 


is equi- 








line DB 
acting ia the direction ps, will produce the same effect or mo- 


tion in a body 8, in that direction, as the oblique force €xpress- 
ed by, and acting in the direction ap, produces in the game di- 
rection cs. And hence any given force as, is to its effect in 
DB, as aB to DB, or as radius to the cosine of the angle asp of 
inclination of those directions. For the same reason, the 
force or effect in the direction az, is to the force or effect in 
the direction ap or rp, as AB to ap; or as radius’ tg, sine of 


the same angle asp, or cosine of the angle paz of those di- 
rections, : : 


ve 
Ba 


43. Corol. 7. Hence also, if the two given forces, to be 
compounded, act in the same line, either both the same way, 
or the one directly opposite to the other ; then their joint or 
compounded force will act in the same line also, and will be 
equal to the sum of the two-when they act the same way, or 
to the difference of them when they act in opposite directions ; 
and the compound force, whether it be the sum or difference, 
will always act in the direction of the greater of the two. 


PROPOSITION. VII. 


44. If Three Forces a, B, c, acting all together in the same Plane, 
keep one another in. Equalibrio ; they will be proportional to 
the Three Sides px, Ec, cv, of a Triangle, which are drawn 
Parallel to the Directions of the Forces ap, DB, cn. 


Propuce ap, Bp, and draw cF, ce, parallel to them. ‘Then 
Poe the 


ie 


: 


120... OF MOTION, FORCES, &c. 


the force in cp is equivalent to the 
two ap, Bp, by the supposition ; but 
the foe “c cp is also equivalent to 
the two ep and ce or Fp; there- 
fore, if cp represent the force C, 
then ep will represent its opposite 
force a, and cE, or FD, its opposite 
force 8. Consequently the three 
forces, A, B, C, are proportional to 
DE, cE, cp, the three lines parallel 
to the directions in which they act. 


45. Corol. 1. Because the three sides cp, cr, pz, are pro- 
portional to the sines of their opposite angles, &,p,c ; there- 
fore the three forces, when in equilibrio, are proportional to 
the sines of the angles of the triangle made of their lines of 
direction ; namely, each force proportional to the sine of the 
angle made by the directions of the other two. 


46. Corol. 2, The three forces, acting against, and keep- 
ing one another i in equilibrio, are also proportional to the sides 
of any other triangle made by drawing lines either perpendi- 
cular to the directions of the forces, or forming any given 
angle with those directions. For such a triangle is always 
similar to the former, which is made by drawing lines parallel 
to the directions ; and therefore their sides are in the same 
proportion to one another. 


47. Corol. 3. If any number of forces be kept in equilibrio 
by their actions agaist one another ; they may be all reduced 
to two equal and opposite ones. Fi or, by cor. 4, prop. 7, any 
two of the forces may be reduced to one force acting in the 
same plane; then this last force and another may likewise be 
reduced to another force acting in their plane ; and go on, till 
at last they all be reduced to the action of only two opposite 
forces : which will be equal, as well as opposite, because the 
whole are in equilibrio by the supposition, 


48. Corol. 4. If one of the 

forces, as c, be a weight, which is ow B ee 
sustained by two strings drawing in 
the directions pa, pp: then the 
force or tension of the string ap, 
is to the weight c, or tension of the 
string pc, as pe to pc; and the 
force or tension of the other string 
Bp, is to the weight c, or tension 
of ¢p, aS cE to cD. . 








49. Corol. 


COLLISION OF BODIES. . 121 
49. Corol. 5. If three forces be in equilibrio by their mu- 
tual actions; the line of direction of each force, as pc, passes 
through the opposite angle c of the parallelogram formed by 
the directions of the other two forces. Sr | 
50. Remark. ‘These properties, in this proposition and its 
corollaries, hold true of all similar forces whatever, whether 
they be instantaneous or continual, or whether they act by 
percussion, drawing, pushing, pressing, or weighing ; and are 
oF the utmost importance in mechanics and the doctrine of 
orces, ee : 


ptr ab 
ON THE COLLISION OF BODIES. 


PROPOSITION IX. 


51. If a Body strike or act Obliquely on a Plain Surface, the 
Force or Energy of the Stroke, or Action, is as the Sine of the 
Angle of Incidence. ei 
Or, the Force on the Surface is to the same uf it had acted Perpen- 
. dicularly, as the Sine of Incidence is to Radius, ~ 


Let as express the direction and 
the absolute quantity of the oblique 
force on the plane be ; or let a given 
body a, moving with a certain velo- 
city, impinge on the plane at B; 
then its force will be to the action 
on the plane, as radius to the sine of Vege 
the angle agp, or as aB, to ap or Bc, drawing ap and gc per- 
pendicular, and ac parallel to pr. _ 

For, by prob. 7, the force az is equivalent to the two forces 
ac, cB; of which the former ac does not act on the plane, 
because it is parallel to it. The plane is therefore only acted 
on by the direct force cs, which is to a8, as the sine of the 
angle Bac, or agp, to radius. rat 

5%. Corol. 1. If a. body act on another, in any direction, 
and by any kind of force, the action of that force on the se- 
cond body, is made only in a direction perpendicular to the 
surface on which it acts. For the force in az acts on pr only 
by the force cs, and in that direction. were 

63. Corol. 2. If the plane pe be not absolutely fixed, it 
will move, after the stroke, in the direction perpendicular to 
its eos For it is in that direction that the force is ex- 
erted, . 


Vou ie oe 17 PROPOSITION 








492 OF MOTION, FORCES, &. 


PROPOSITION X. 
5A. if one Body a, strike another Body B, which is either at Rest 
or moving towards the Body a, or moving from it, but with a 
less Velocity than that of a, then the Momenta, or Quantities of 


| Motion, of the two Bodies, estimated in any one Direction, 
_ will be the very same after the Stroke that they were before it. 


For, because action and re-action are always equal, and in. 
contrary directions, whatever momentum the one body gains 
one way by the stroke, the other must just lose as much in 
the same direction ; and therefore the quantity of motion in 
that direction, resulting from the motions of both the bodies: 
remains still the same as it was before the stroke. 


55. Thus, if a with a momentum 
of 10, strike pn at rest, andcommu- @———-O———— 
nicate to ita momentum of 4,inthe A. Bi) Cc 
direction as. Then a will have 
only a momentum of 6 in that direction ; which, together with 
the momentum of 8, viz. 4, make up still the same momentum 
between them as before, namely, 10. 


56. If 8 were in motion before the stroke with a momen- 
tum of 5, in the same direction, and receive from a an addi- 
tional momentum of 2. Then the motion of a after the stroke 
will be 8, and that of 8, 7; which between them make 15, 
the same as 10 and 5, the motions before the stroke. s 


57. Lastly, if the bodies move in opposite directions, and 
meet one another, namely, a with a motion of 10, and B, of 
5; and A communicate to B a motion of 6 in the direction aB 
of its motion. Then, before the stroke, the whole motion 
from both, in the direction of ab, is 10—5or5. But, after 
the stroke, the motion of ais 4 in the direction aB, and the 
motion of g is 6—5 or | in the same direction aB ; therefore 
the sum 4 + 1, or 5, is 8 still the same motion from both as it 
was: ‘before. 


PROPOSITION XI. 


58. The Motion of Roihits ductuded @n' a Given Space, is the 
same with regard to each other, whether that Space be at Rest, 
‘or move uniformly ¢ ina Right Line. 
Mon ie any force be equally impressed both. on the body 
and the line in which it moves, this will cause no chanky > in 


the 


COLLISION OF BODIES. 428 


the motion of the body along the right line. For the same 
reason, the motions of all the other bodies, in their several 
directions, will still remain the same. Consequently their 
motions among themselves will continue the same, whether 
the ineluding space be at rest, or be moved uniformly for- 
ward. And therefore their mutual actions on one aagihcr, 
must algo remain the same in both cases. | 


PROPOSITION XII. _ 


59. ‘If a Hard and Fixed Plane be struck by either a Soft ore 
- Hard Unelastic Body, the Body will continue onit.  Butif the 
Plane be struck by a Perfectly Elastic Body, it will rebound 
from tt again with the same Velocity with which it struck the 
Plane. 


- For, since the parts which are struck, of the elastic body, 
suddenly yield and give way by the force of the blow, and as 
suddenly restore themselves again with a force equal to the 
force which impressed them, by the definition of elastic bodies ; 
the intensity of the action of that restori ing force on the plane, 
will be equal to the force or momentum with which the body 
struck the plane. And, as action and re-action are equal and 
contrary, the plane will act with the same force on the body, 
and so cause it to rebound or move back again with the same 
velocity as it had before the stroke. 

But hard or soft bodies, being devoid of elasticity, by the 
definition, having no restoring force to throw them off again, 
they must necessarily adhere to the plane struck. 


60. Corol. 1. The effect of the blow of the elastic body, 
on the plane, is double to that of the unelastic one, the velo- 
city and mass being equal in each. 

For the force of the blow from the unelastic body 18 as its 
mass and velocity, which is only destroyed by the resistance of 
the plane. But in the elastic body, that force is not only de- 
stroyed and sustained by the plane ; but another also equal to 
it is sustained by the plane, in consequence of the restoring 
force, and by virtue of which the body is thrown back again 
with an equal velocity. And therefore the intensity of the blow 
is doubled. ! 


61. Corol. 9, Hence rinelatic bodies lose, by their collision, 
only half the motion lost by elastic bodies ; their mass and 
velocities being equal. For the latter communicate double 


the motion of the former. 
PROPOSITION 


24 OF MOTION; FORCES, we. 
‘PROPOSITION XIII. aka 


62. If an Elastic Body a impinge on a Firm Plane ve at thé 
Point w, it will rebound from tt in an Angle equal. to that in 
which it struck it; or the Angle of Incidence will be equal to 
the Angle of Reflection; namely, the Angle asp equal to the 
Angle FBE. ni a 


_ Ler as. express the force of 
the body ain the direction as ; 
which let be resolved into the 
two ac, cB, parallel and per- 
pendicalar to the plane.—Take 
BE and cr equal to ac, and draw 
BF. Now action and re-action being’ e€ 
resist the direct force cs by another Bc equal to it, and in a 
‘contrary direction: whereas the other ac, being parallel to 
the plane, is not acted on or diminished by it, but stil] con- 
tinues as before. The body is therefore reflected from the 
plane by two forces.sc, BE, perpendicular and parallel) to the 
_ plane, and therefore moves m the diagonal sr by composition. 
But, because ac is equal to Be or cr, and that Bc is common, 
the two triangles Bc, sor are mutually similar and equal ; and 
consequently the angles at a and F are equal, as also their 
equal alternate angles asp, rBx, which are the angles of inci- 
dence and reflection. ‘i : 

















PROPOSITION XIV. 


63. To determine the Motion of Non-elastec Bodies when they 
- strike each other Directly, or an the same Line of Direction. 


‘Lert the non-elastic body z, mov- ' 
ing with the velocity vinthe di- @_. O 2 
rection bb, and the body 6 with 7p 6b C 
the velocity v, strike each other. cage 
Then because the momentum of any moving body is as the 
mass into the velocity, sv = m is the momentum of the body, 
Bp, and bv = mthe momentum of the body 6, which let be 
the less powerful of the two motions. ‘Then, by prop. 10, 
the bodies will both move together as one mass in,the direc- 
tion sc after the stroke, whether before the stroke the body 
6 moved towards c or towards's. Now, according as that 
motion of 6 was from or towards 8, that is whether: the 
motions were in the same or contrary ways, the momentum 
after the stroke, in direction gc, will be the sum of difference 

sic eatape of 


COLLUSION OF BODIES. 125 


of the momentums before the stroke ; namely, the momen- 
tum in direction Bc will be 203 


ee if the bodies moved the same way, or 
pv —dv, if they moved contrary ways, and 
BV only, if the body 6 were at rest.’ . 


Then divide each momentum by the common mass of mat-- 
ter + b, and the quotient will be the common velocity after 
the stroke in the direction ze ; namely, the common velocity 
will be, in the first case, 


BYV+bv . BV — bv BU 


Bho” inthe 2d ae and in the 3d zat 


64. For example, if the bodies, or weights, B and 6, be as 
5 to 3 and their velocities v and v, as 6 to 4, oras 3 to 2, be- 
_ fore the stroke ; then 15 and 6 will be as their momentums, 
aad 8 the sum of their weights ; consequently, after the 
stroke, the common velocity will be as. 


15+6 21 i, | 
sar ede or 2% in the first case, 








8 
15—6 9 
oct 5 or 11 in the second, and 
15 3 
eeu dialaes or 12 in the third. 


_ PROPOSITION XV. * 


55. If two Perfectly Elastic Bodies impinge on one another : 
their Relative Velocity will be the same both Before and After 

the Impulse: that is, they will recede from each other with the 
_ same Velocity with which they approached and met. 


For the compressing force is as the intensity of the stroke ; 
which, in given bodies, is as the relative velocity with which 
they meet or strike. But perfectly elastic bodies restore 
themselves to their former figure, by the same force by which 
they were compressed ; that is the restoring force is equal to 
the compressing force, or to the force with which the bodies 
approach each other before the impulse. But the bodies are 
impelledfrom each other by this restoring force ; and there- 
fore this force, acting on the same bodies, will produce a re-. 
lative velocity equal to that which they had before: or it will 
make the bodies recede from each other with the same velo- 

city 


126 OF MOTION, FORCES, & 


city with which they before approached, or so as to be equally 
distant from one another at equal times before and after the 
impact. 

66. Remark. it is not meant by this froposition, that each 
body will have the same velocity after the impulse as it had 
before ; for that will be varied according to the relation of 
the masses of the two bodies; but that the velocity of the 
one will be, after the stroke, as much increased as tbat of the 
other is decreased, in one and the same direction. So, if the 

- elastic body B move with a velocity v, and evertake the elas- 
tic body 6 moving the same way with the velocity v; then 
their relative velocity, or that with which they strike, is v—v, 
and it is with this same velocity that they separate from each 
other after the stroke. But if they meet each other, or the 
body 6 move contrary to the body 8; then they meet and 
strike with the velocity v--v, and it is with the same velocity 
that they separate and recede from each other after the stroke. 
But whether they move forward on backward after the im- 
pulse, and with what particular velocities, are circumstances 
that depend on the various masses and velocities of the bo- 
dies before the stroke, and which make the subject of the 
next proposition. 


PROPOSITION XVI. 


67. To determine the Motions of Elastic Bodies after Striking 
each other directly. 


Let the elastic body 8 move in 3 O 
the direction sc, with the velocity b C 


v,; and let the velocity of the other 

body 6 be v in the same line ; which latter velocity v will be 
positive if b move the same way as 8, but negative if 6 move 
in the opposite direction to B. Then their relative velocity 
in the direction sc is v—‘v; also the momenta before the 
stroke are sv and bv, the sum of which 1 is BV + 5v in the di- 
rection BC. 

Again, put x for the velocity of s, and y for that of 5, in 
the same direction sc, after the stroke ; then their relative 
velocity is y—, and the sum of their momenta Bx +-dy in the 
game direction. 

But the momenta before and after the collision, estimated 
in the same direction, are equal, by prop. 10, as also the re- 
lative velocities, by the last prop. Whenee arise these two 
‘inal 

Fiz: 


COLLISION OF BODIES. 127 


viz. By-+-bv=s2x-+by, 
-and v— v= Y—25 - 
the resolution of which equations gives 





+2bv 
Fok ah apt fare the velocity of B, 
y= a ee —--, the velocity of b, 


both in the direction sc, when v and 2 are both positive, or 
the bodies both moved towards c before the collision. But if 
v be negative, or the body 6 moved in the contrary direction 
before collision, or towards 8; then, changing the sign of », 
the same theorems become 








(p—b) v—2bv sae 
x ea , the velocity of 8, 
y= at ae the veloc. of b, in the direction sc. 


And if 6 were at rest before the impact, making its velecity 
»=0, the same theorems give 
B=—6 28 
lage ear pe and y =--———— v, the velocities in this case. 


B-++B 

And in this case, if the ate bodies zs and 5 be equal to each 

2p : 

h wie | 2B 2B pe 
other; then s—b = 0, and ay on 1; which give z=0, 
and y==v; that is the body 8 will stand still, and the other 
body 6 will move on with the whole velocity of the former : 
a thing which we sometimes see happen in playing at billiards ; 


and which would nae much oftener if the balls were per- 
fectly elastic. 


PROPOSITION XVII. 


88. If Bodies strike one anotker Obliquely, it 1s proposed to de- 
termine their Motions after the Stroke. 


Let the two bodies B, 6, 
move in the oblique directions 
Ba, ba, and strike each other 
at a, with velocities which are 
in proportion to the lines Ba, 
ba; to find their motions after 
the impact. Let can repre- 
sent the plane in which the 
bodies touch in the point of 
concourse ; to which draw the perpendiculars Bc, bp, and 
complete the rectangles cz, pr. Thenthe motion in Ba is re-. 

solved 





128 OF MOTION, FORCES, &e. 


solved into the two gc, ca; and the motion in ba is resolved 
‘into the two bp, pa; of which the antecedents sc, bp, are the 
velocities with which they directly meet, and the congequents 
CA, DA, are parallel; therefore by these the bodies do not im- 
pinge on each other, and consequently the motions, according 
te these directions, will not be changed by the impulse; so 
that the velocities with which the bodies meet, are as gc and 
bp, or their equals rs and ra. The motions therefore of the 
bodies g, b, directly striking each other with the velocities za, 
Fa, will be determined by prop. 16 or 14, according as the 
bodies are elastic or non-elastic ; which being’ done, let ac be 
the velocity, so determined, of one of them, as B; ‘and since 
there remains also in the body a force of moving in the direc- 
tien parallel to se, with a velocity as BE, make an equal to 
BE, and complete the rectangle cH: then the two motions in 
AH and AG, or HI, are compounded into the diagonal a1, which 
therefore will be the path and velocity of the body 8 after the 
stroke. And after the same manner is the motion of the other 
body 4 determined after the impact. . 

If the elasticity of the bodies be imperfect in any given de- 
gree, then the quantity of the corresponding lines must be di- 
minished in the same proportion. 


cee cee: 


THE LAWS OF GRAVITY ; THE DESCENT OF HEAVY 
BODIES; AND THE MOTION OF PROJECTILES IN 
FREE SPACE. — 


4; 


PROPOSITION XVIII. 
69. All the properties of Motion delivered en Proposttion VI, tts 


Corollaries and Scholium, for Constant Forces, are true in the 
Motions of Bodies freely descending by ther own Gravity ; 
namely, that the velocities are as the “Times, and the Spaces as 
_ the Squares of the Times, or as the Squares of the Velocities, 


For, since the force of gravity is uniform, and constantly 
the same, at all places near the earth’s surface, or at nearly 
the same distance from the centre of the earth's ; and since 
this is the force by which bedies descend to the surface ; they 
therefore descend by a force which acts constantly and equal- 
ly ; consequently all the motions freely produced by gravity 
are as above specified, by that proposition, &c. 


“ SCHOLIUM. 


70, Nore it has been found, by numerous experiments, 
that 


“OF eravity. fy 


that gravity is a force of such a nature, that all bodies, whether 
light or heavy, fall perpendicularly through equal spaces in 
the same time, abstracting from the resistance of the air; yas 
lead, or gold, and a feather, which in an exhausted receiver 
fall from the top to the bottom in the same time. It is also 
found that the velocities acquired by descending, are in the 
exact proportion of the times of descent: and further, that 
the spaces descended are proportional to the squares of the 
times, and therefore to the squares of the velocities. Hence 
then it follows, that the weights of gravities, of bodies near 
the surface of the earth, are proportional to the quantities of 
matter contained in them ;-and that the spaces, times, and 
velocities, generated by gravity, have the relations contained 
in the three general proportions before laid down. Further, 
as it is found, by accurate experiments, that a body in the 
latitude of London, falls nearly 163, feet in the first second 
of time, and consequently that at the. end of that time it has 
acquired a velocity double, or of 32} feet by corol. 1, prop. 6 ; 
therefore if g denote 16), feet, the space fallen through i in 
one second of time, or Oe ine velocity generated in that time ; 
then, because the velocities are directly proportional to the 
times, and the spaces to the “sarin of the times ; therefore 
it will be, 
as 1": t’:: 2g : Q99t =v the velacity, 
and 12:22 :: g: gt?= s the space. , 

SS6 that, for the descents of gravity, we have these general 

eq namely, 


¥ v2 
$= gi? ages = liv. 
2s 
v= 2gt repre = 2/85. 
ate v Qs gene 
ee we 
v 26 v? 


Hence, because the times are as the. valseitten: and the 
spaces as the squares of either, therefore, c: | 

if the times be as the numbs. 1, 2,3, 4, 5, &c. 

the velocities will also be.as 1, 2. 3, 4p% 5; Sess 

and the spaces as their squares i 4, 9,16, 25, &c. 

and the space of each time as 1, 3,5, 7, 9, &c. 
namely, as the series of the odd numbers; whith are the 
differences of the squares denoting the whole spaces. So 
_ that if the first series of natural numbers be seconds of time, 
Vor. JT, . 18 namely, 


130 OF MOTION, FORCES, &c. 


namely, the times in seconds 1”, 2”, = 8", 4”, &c. 
the velocities in feet will be 323, 644, 961, 1282, &c. 
the spaces in the whole times. 164, 641 , 1448, 2592, Se. 
and the space for each second . 1635, 481 , 80,5, 1127, &e. 
Ti. These relations, of the times, veloci- A 
ties, and spaces, may be aptly represented 
by certain lines and geometrical figures. ad 
Thus, if the line an denote the time of any } € 
body’s descent, and sc, at right angles to it, 
ihe velocity, gained at the end of that time 30-5 
by joining ac, and dividing the time aB into B : Cc 
any number of parts at the points a, b, c; 
then shall ad, be, ef, parallel to Bc, be the velocities at the 
ne of time a, b, c, or at the ends of the times, aa, ab, 
; because these latter lines, by similar triangles are pro- 
i ttiganl to the former ad, be, cf, and the times are propor- 
tional to the velovities. Also, the area of the triangle apc 
will represent the space descended through by the force of 
gravity in the time as, in which it generates the velocity zc ; 
because that area is equal to tan Xzc, and the space descend- 
edis. s =1tv. or half the product of the time and the last 
velocity. ” And, for the same reason, the less triangles aad, 
Abe, acf, will represent the several spaces described in the 
corresponding times aa, ab, ac, and-velocities ad, be, ef; those 
triangles or spaces being also as the squares of their like sides 
aa, ab, ac, which represent the times, or of ad, al fs which 
represent ‘the velocities. 


72. But as areas are rather unnatural P q@ % e f 
representations of the spaces passed over : 
by a body in motion, which are lines, the 
relations may better be represented by 
the abscisses and ordinates of a parabola. 
Thus, if pa be a parabola, pr its axis, 
and rq its ordinate; and pa, pb, rc, &c. 
parallel to ra, represent the times from & Q 
the beginning, or the velocities, then ae, bf, eg, &c. parallel - 
to the azis pr, will represent the spaces described by a fall- 
‘ing body in those times ; for, ina parabola, the abscisses ph, 
pt, Pk, &c. or ae, bf, eg, &c. which are the spaces described, 
are as the squares of the ordinates, he, if, kg, &c. or ra, Pb, 

) PC, &c. which represent the times or velocities. 





a3. And because the laws for the destruction of motion, 
. “> are 


OF GRAVITY. — 131 


are the same as those for the generation of it, by la forces, 
but acting in a contrary direction ; therefore, 
Ist, A body thrown directly inward with any peladly will 
lose equal velocities in equal times. 
2d, If a body be projected upward, with the velocity it 
acquired in any time by descending freely, it will lose all its 
velocity in an equal time, and will ascend just to the same 
height from which it fell, and will describe equal spaces in 
equal times, in rising and falling, but in an inverse order ; and 
it will have equal velocities at any one and the same point of 
the line described, both in ascending and descending. 
3d, If bodies be projected upward, with any velocities, 
the height ascended to, will be as the squares of those veloci- 
ties, or as the squares of the times of ascending, till they lose 
all their velocities. 
74. To illustrate now the rules for the natural descent of 
bodies by a few examples, let it be required. | 
‘Ast, ‘To find the space descended ty a body in 7 seconds of 
time, ‘and the velocity acquired. 
Ans. 7883, space ; and 225% elocity. 
2d, To find the time of eenerating a velocity of 100 feet 
per second, and the whole space sa palaces 
Ans. 3°23, time ; 155,33, space. 
3d, To find the time of descending 400 feet, and the velo- 
city at the end of that time. 
Ans. 4°28 time ; and 16033 a yeleeyy: 


PROPOSITION XIX. 


75. Ifa Body be projected in Free Space either Parallel to the 
Horizon, or in an Oblique Direction, by the Force of Gun- 
Powder, or any other Impulse ; rt will by this Motion, in 
- Conjunction with the Action of Gravity describe the Curve 
‘Tine of a Parabola. 





AS 
Ler the body be projected from the point a, in the direc- 


tion ap, with any uniform velocity : then, in any equal por- 
tions 


132 OF MOTION, FORCES, &c. 


tions of time, it would by prop. 4, describe the equal spaces 
AB, BC, cp, &c. in the line ap, if it were not drawn continu- 
ally down below that line by the action of gravity. Draw Bz, 
oF, pc, &c. in the direction of gravity, or perpendicular to 
the horizon, and equal to the spaces through which the body 
would descend by its gravity in the same ‘time in which it would 
uniformly pass over the corresponding spaces az, ac, ap, &c. 

by the projectile motion. Then, since by these two motions 
the body is carried over the space an, in the same time as 
over the space BE, and the space ac in the same time as the 
space cr, and the space ap in the same time as the space pc, 
_&e. ; therefore, by the composition of motions, at the end of 
“those times, the body will be found respectively i in the points 
E, F, Gc, &c.; and consequently the real path of the projectile 
will be the curve line abFG, &c. But the spaces as, ac, av, 
&c. described by untform motion, are as the times of descrip- 
tion ; and the spaces Br, cr, pe, &c. described inthe same times 
by the accelerating force of gravity, are as the squares of the 
times ; consequently the perpendicular descents are as the 
squareseof the ‘spaces in ap, that is BE, CF, DG, &c. are res- 
pectively proportional to ant, 4c2, AD?, &e. ; which is the 
property of the parabola by theor. 8, Con. Sect. Therefore 
the path of the projectile is the parabolic line arra, &c. to 
which ap is a tangent at the point a. 


76. Corol. 1. The horizontal: velocity of a projectile, is 
always the same constant quantity, in every point of the 
curve ; because the horizontal motion is in a constant ratio 
to the motion in av, which is the uniform projectile motion. 
And the projectile velocity is in proportion to the constant 
horizontal velocity, as radius to the cosine of the angle pam, 
or angle of elevation or depression of the piece above or he 
low the horizontal line au. 


. Corol. 2, ‘The velocity of the projectile i in the direc- 
i Of the curve, or of its tangent at any point a is as the se- 
cant of its angle Bar of direction above the horizon. For the, 
motion in the horizontal direction a1 is constant, and a1 is to 
AB, as radius to the secant of the angle a; therefore the mo- 
tion at A, in AB, is every where as the secant of the angle A. 


78. Corol. 3.. The velocity in the direction pe of gravity 
or perpendicular to the horizon, at any point ¢ of the curve, 
is to the first uniform projectile velocity at a, or point of 
contact of a tangent, as 2cp isto ap. For, the times In AD 
and DG @ being equal, and the velocity acquired by freely de- 

scending 


‘PROJECTILES. 133 


scending through pe, being such as would carry the body uni- 
formly over. twice De in an equal time, and the:spaces describ- 
ed with uniform motions being as the velocities, therefore the 
space ap is to the space 2pe, as the projectile velocity. at a, 
to the perpendicular velocity at c. 


PROPOSITION XX. 


‘79. The Velocity in the Direction of the Curve, at any Point of 
it as a, ts equal to that which is generated by Gravity in free- 
ly descending through a Space which is equal to One-Fourth 


of the Parameter of the diameter of the Parabola at. that 
Point. . 


Let pa or ab be the heightdue 
to the velocity of the projectile 
at any point a, in the direction of 
the curve or tangent ac, or the ve- — 
locity acquired by falling through 
that height; and complete the 
parallelogram acpsp. ‘Then is cp 
==as or ap, the height due to the 
velocity in the curve at a and cp Is also the height due to the 
perpendicular velocity at p, which must be equal to the former; 
but by the last corol. the velocity at ais to the perpendicular 
velocity at p, as ac to 2cp ; and as these velocities are equal, 
therefore ac or Bp is equal to 2cp, or 2an; and hence ap or 
ap is equal to 18p, or } of the parameter of the diameter AB, 


by corol. to theor. 13 of the Parabola. : 





80. Corol. 1. Hence, and from cor. 2, yy er yy aT 
theor. 13 of the parabola, it appears that, , “7-777 fap ; 
if from the directrix of the parabola ak | 
which is the path of the projectile, seve- © |/H i 
ral lines-He be drawn perpendicular to a \ 
the directrix, or parallel to the axis; then nd 
the Velocity of the projectile in the direction of the curve, 
at any point &, is always equal to the velocity acquired by a 
body falling freely through the perpendicular line un, 

8t. Corol. 2. If a body, after falling through the height pa 
(last fig. but one), which is equal to a8; and when it arrives 
at A, have its course changed, by reflection from an elastic 
plane at, or otherwise, into any direction ac, without altering 
the velocity ; and if ac be taken = 2ap or 2ap, and the paral- 


lelogram be completed ; then the body will describe the pa- 
rabola passing through the point p. 





82. Corol. 


134 OF MOTION, FORCES, &c. 


. Corol. 3. Because ac = 248 or 2cp or 2apP, therefore 
ies 2ap X2cp or ap. 4cp.; and because all the perpendicu- 
lats EF, CD, GH, are as AE?, ac, aG?; therefore alse ap . 4rF 
== an?, and ar . 460 = ‘act, &e. ; and because the rectangle 
of the extremes is equal to the rectangle of the means of four 
proportionals, therefore always 


Gs pes > AE: : AB: 4EF,. 
and ap : ac i: ac: 4cp, 
and AP : AG :: aG : 4GH,. 
and so on. 


PROPOSISION XXI. 


83. Having gvven the Direction, and the Impetus, or Altitude 
due to the First Velocity of a Projectile ; to determine the 
' Greatest Height to which it will rise, and the Random or Ho- 
rizontal Range. 


Ler ap be the height due to the ae See A 
projectile velocity at a, ac the di- ; 
rection, and au the borizon.. On G L hey 
; i ; ae 
aG let fall the perpendicular re, a | 
and on ap the perpendicular ar; N\oZ iv . 








so shall az be equal to the greatest ew: | Pa 
altitude cv, and 4ar equal to the 4 — 2h 
A C H 
horizontal range aH. Or, having 
praise pq perp. to ac, take ac == 4aa, and draw cH perp. to 
- then au is the range. 


fa. hy the last corollary, , ap: aG@:: sc: 4cn; 
and, by similar triangles, APiD AGES? BQ UGH, 
OF Hc pe ES) Ae > 4aQ: 46H; 


therefore ac=4@ ; and, by Cates fvisninlen, An=4QR. 
Also, if v ye Re vertex of the arch ree AB OF 7AG= 


NAL 


property of | the Pa sia 
84. Corol. 1. Because the angle 
g is aright angle, which is the angle 
wma semicircle, therefore if, on ap 
as a diameter, a semicircle be de- 
scribed, it will pass through he 
ol 
ees Corol, 2. Ifthe horizontal range 
and the peeneenr velocity be giv the : 
the 





- PROJ ECTILES. . 13h 


the direction of the piece so as to hit the object x, will be 
thus easily found : Take aD =1 aH, draw DQ perpendicular 
to AH, meeting the semicircle, described on the diameter ap, 
ing and q: then. AQ or aq will be the direction of the piece. 
And hence it appears, that there are two directions ax, ax, 
which, with the same projectile velocity, give the very same 
horizontal range au. And these two directions make equal 
angles qap, QAP with au and ap, because the arc Pe =the arc 
Aq. oh 

86. Cerol. 3. Or, if the canie AH, ms Hipcotan: AB, be giv- 

n; to find the altitude and velocity or impetus. Take ap= 
Jan, and erect the perpendicular pe, meeting ap in Q ; so 
shall pe be equal to the greatest altitude cv. Also, erect ap 
perpendicular to an, and ap to ag; so shall ap be the height 
due to the velocity. 


87. Corol. 4. When the body i is projected with the same 
velocity, but in different directions : the horizontal ranges 
au will be as the sines of double the angles of elevation.— 
Or, which is the same, as the rectangle of the sine and co- 
sine of elevation. For ap or ra, which is 1a4, is the sine 
of the arc ae, which measures double the angle qap of ele- 
vation. 

And wken the direction is the same, but the velocities diffe- 
rent; the horizontal ranges are as the square of the veloci- 
ties, or as the height ar, which is as the square of the veloci- 
ty; for the sine ap or r@ or janis as the radius or as the 
diameter ap. 

Therefore, when both are different, the ranges are in ‘the 
compound ratio of the squares of the velocities, and the sines 
of double the ane of elevation. 


88. Corol. 5. The greatest range. is when the angle of ele- 
vation is 45°, 0 or half a right angle; for the double of 45 is 
90, which has the greatest sine. Or the radius os, which is 
4, of the range, is the greatest sine. 

_ And hence the greatest range, or that at an elevation of 45° 
is just double the altitude ap which is due to the velocity, or 
equal to 4vc. Consequently, in that case, c is the focus of 
the parabola, and an its parameter. Also, the ranges are 
equal, at angles equally above and below 45°. 

89. Corol. 6. When the elevation is 15°, the double of 
which, or 30°, has its sine equal to half the radius ; conse- 
quently then its range will be equal to ap, or half the greatest 
range at the elevation of 45°; that is, the range at 15°, is 
equal to the impetus or height due to the projectile ee 

90. Corol. 


136 OF MOTION, pees: Ke. 


90. Corol. 7. The greatest altitude cv, being equal to ar, 
is. as the versed sine of double the angle of elevation, and 
also as ap or the square of the’ velocity. Or as the square 
of the sine of elevation, and the square of the velocity ; for 
- the square of the sine is as the versed sine of the double” an- 

gle. : ; % B 


91. Corol. 8. The time of flight of the projectile; Phish: is 
equal to the time of a body falling freely through cH or 
4cv, four times the altitude, is therefore as the square root 
of the altitude, or as the projectile velocity and'sine of the 
élevation. 


‘SCHOLIUM. 


92. From the last proposition, and its corollaries, may be 
deduced the following set of theorems, for finding all the cir- 
cumstances of projectiles on horizontal planes, having any 
two of them given. ‘Thus, let CEO denote the sine, cosine, 
and tangent of elev ation ; ;_ 8; a the sine and versed sine of 
the double elevation ; rx the horizontal range ; T the time of 
flight ; v the projectile velocity ; ; H the greatest height of the 
projectile g = 16,1, feet, and a the impetus, or the altitude 
due to the velocity. V. Then, 





Sy? ¢ oe Sey? cer? T? 4H 
Rose Das = | Angee, pee Fd _gvt_ ty ane i 
% 2g g 3 s t t 
oR gR gT g 
y¥ =,/log=,/—- = ./— — = — H 
v v s v SC s vs 
Eee oes y tr sr aye 
PS SS, f/f — = ~~ = 22. 
& & i & Es AK 
: SR... $207 vv? g 
a= at= lw ln > — $= —_ = = > = 


And from any of these, the angle of direction may be found. 
Also, in these theorems, g a in many cases, be taken = 16, 
without the small fraction 7} pn reek will be n near enough for 


common use. 
Ay PROPOSITION XXII. 


93. To determine the Range on an Oblique Plane ; Lbs 
| a jaa the Impetus or Veloenty, and the ee a Direction. 


‘Ler ar be the oblique plane, at a given angle, iter 
above or below the horizontal plane | AH; AG the direction 


PROJECTILES. 137 


of the piece, and ap the alti- 
tude due to the projectile ve- 
locity at a. 

By the last proposition, find 
the horizontal range aH to 
the given velocity and direc- 
tion ; draw HE perpendicular 
to AH, meeting the oblique 
plane in 8; draw er parallel 
to ac, and ri parallel to ue ; 
go shall the projectile pass through t, and the range on the 
oblique plane will be ar.. As is evident by theor. ‘15 of the 
Parabola, where it is proved, that if au, at be any two lines 
terminated at the curve, and ir, He parallel to the axis; then 
is EF parallel to the tangent ac. 





94. Otherwise, without the Horizontal Range. 


_ Draw re perp. to aa, and ap perp. to the horizontal plane 
aF, meeting the inclined plane ink; take ar = 4axk, draw EF 
parallel to ac, and Fi parailel to ap Or pe; so shall ar be the 
range on the ‘oblique plane. For an =4 ap, therefore zu is 
parallel to rr, and so on, as above. 


Otherwise. 


95. Draw pq making the angle apg = the angle car; then 
take ac = 4aq, and draw at perp. to an. Or, draw ght perp. 
to aH, and take ar = 4ak. Also xq will be ee to cv the 
greatest height above the plane. 

For, by cor. 2, prop. 20, AP: aG:: AG: 4cI 

and by sim. triangles, APB 2:AG)$ 5° °ag 2” GE, 

2 Te tate Re ee Oe > 4aq : 4c1; 
therefore ac = 4aq; and by sim. “trisablel ars=4ak. 

Also gk, or 1 a1, is = to cv by theor. 13 of the Parabola. 





96. Corol. 1. If ao be drawn perp. to the plane ar, and 
Von. I. 19 ap be 


138 OF MOTION, FORCES, &c. 


ap be bisected by the perpendicular sro; then with the cen- 
tre o describing a circle through a and Pp, the same will also 
pass through q, because the angle car, formed by the tangent 
Aland ac, is equal to the angle arg, which will therefore 
stand on the same arc ag. 


97. Corol. 2. If there be given the range ar and the velo- 
city, or the impetus, the direction will hence be easily found 
thus: Take ak=1a1, draw kq perp. to an, meeting the circle 
described with the radius ao in two points g and q; then ag or 
‘ag willbe the direction of the piece. And hence it appears: 
that there are two directions, which, withthe same impetus, 
give the very same range a1. And these two directions make 
equal angles with a1, and ap, because the arc pg is equal the 
arc 4g. They also make equal angles with a line drawn from. , 
a through s. because the arc sq is equal the arc sq. 


98. Corol. 3. Or, if there be given the. range a1, and the. 
direction ag; to find the velocity or impetus. Take ak = 
2 at, and erect kg perp. to au, meeting the line of direction. 
in g; then draw ge making the Z age = Z akq; so shall ar 
be the impetus, or the altitude due to the projectile velocity. 


99. Corol. 4. The range on an oblique plane. with a given. 
elevation, is directly proportional to the rectangle of the co- 
sine of the direction of the piece above the horizon, and the- 
sine of the direction above the oblique plane, and reciprocal- 
ly to the square of the cosine of the angle of the plane above. 
or below tke horizon. 


For, put s = sin. Z.gAI or apg, 
¢ = cos. ZgAH OF Sin. PAY, 
c = cos. Z14H or sin. akd or akg or agp. 


Then, in the triangle APG, @ 3.63 "SAPit Ags 
and in the triangle akg = : 2 : Ag : Ak; 
theref. by composition, c?:cs:: ap: aK=Hal. 


So that the oblique range at = x 4ap. 
Cc? 


100. The range is the greatest when ak is the greatest ; 
that is, when kq touches the circle in the middle points; 
and then the line of direction. passes through s, and bisects. 
the angle formed by the oblique plane and the vertex. Also, 
the ranges are equal at equal. angles above and below this di- 
rection for the maximum. 

101.: Corol. 5..'The greatest height cv or kg of the projec- 

| tile, 


~~ PROJECTILES. 139 


he se . 2 
tile, above the plane, is equal to “ * ap. And therefore it 


is as the impetus and square of the sine of direction above 
the plane directly, and square of the cosine > of fy ae s in- 
clination reciprocally. 
For - c (sin. agp) : s (sin. apg) :: AP: ag, 
and c (sin. akg) :s en Ba) ag : kg, 
theref. by comp. c? : s?: : ap : kq. 
102. Corol. 6. The time of flight in the curve avi is = 


==, where g = 164, feet. And therefore it is as the 


elgcity and sine of direction above the plane directly, and. 
cosine of the plane’s inclination reciprocally. For the time 
of describing the curve, is equal to the time of falling freely 


433 

through ei or 4kq or —X ar. Therefore, the time being 
c4 

as the square root of the distance, 


2 » 
WV i JAP ue: eve, the time of flight. 


SCHOLIUM. | 


103. From the foregoing corollaries may be collected the 
following set of theorems relating to projects made on any 
given inclined planes, either above or below the horizontal 
plane. In which the letters denote as before, namely, 


c = cos. of direction above the horizon, 

c = cos. of inclination of the plane, 

$ = sin. of direction above the plane, 

the range on the oblique plane, © 

the time of flight, 

the projectile velocity, 

the greatest height above the plane, 

the impetus, or alt. due to the velocity v, 
g = 163, feet. Then, 


Re<4 Bw 





ge 4c 
=— X4a= — v? ==-T? >—H 
2 s $ 
s? $s? y2 SR 
Ea | =— a5 p2 
3 4gc? 4c 
4 ge 2c 
v= ag =¢,/— =-——T =—— 1/2 
A/ 85 s savor 
Os) -a:..6V. SR. H 
Yess Je. St = f/—= Q ri de 
c c gc 


And from any of these, the angle of direction may be found. 
PRAC- 


140 OF MOTION, FORCES, &c. 


PRACTICAL GUNNERY. 


104. THE two foregoing propositions contain the whole 
theory of projectiles, with theorems for all the cases, regu- 
larly arranged for use, both for oblique and horizontal planes. 
But, before they can be applied to use in resolving the seve- 
ral cases in the practice of gunnery, it is necessary that some 
more data be laid down, as derived from good experiments 
made with balls or shells discharged from cannon or mortars, 
by gunpowder, under different circumstances. For, without 
such experiments and data, those theorems can be of very 
little use in real practice, on account of the imperfections and 
irregularities in the firing of gunpowder, and the expulsion of 
balls from guns, but more especially on account of the enor- 
mous resistance of the air to all projectiles made with any 
velocities that are considerable. As to the cases in which 
projectiles are made with small-velocities, or such as do not 
exceed 200, or 300, or 400 feet per second of time, they may 
be resolved tolerably near the truth, especially for the larger 
shells, by the parabolic theory, laid down above. But, in 
cases of great projectile velocities, that theory is quite inade- 
quate, without the aid of several-data drawn from many and 
good experiments. Jor so great is the effect of the resistance 
of the air to projectiles of considerable velocity, that some of 
those which in the air range only between 2 and 3 miles at 
the most, would in vacuo range about ten times as far, or be- 
tween 20 and 30 miles. 

The.effects of this resistance are also various, according to 
the velocity, the diameter, and the weight of the projectile. 
So that the experiments made with one size of ball or shell, 
will not serve for another size, though the velocity should be 
the same ; neither will the experiments made witb one ve- 
locity, serve for other velocities, though the ball be the same. 
And therefore it is plain that, to form proper rules for practi- 
cal gunnery, we ought to have good experiments made with 
each size of mortar, rand with every variety of charge, from 
the least to the greatest. And not only so, but these ought 
also to be repeated at many different angles of elevation, 
namely for every single degree between 30° rand 60° elevation, 
and at intervals of 5° above 60° and below 30, from the ver- 
tical direction to point blank. By such a course of experi- 
ments it will be found, that the greatest range, instead of be- 
ing constantly that at an elevation of 45°, as in the parabolic. 
theory, will be at all intermediate degrees between 45 and 30 ; 

being 


‘PRACTICAL GUNNERY. it 


being more or less, both according to the velocity and the 
weight of the projectile ; the smaller velocities and larger 
shells ranging farthest when projected almost at an elevation 
of 45°; while the greatest velocities, especially with the 
smaller shells, renge farthest with:an elevation of about 30°. 

105. There have, at different times, been made certain 
small parts of such a course of experiments as is hinted at 
above. Such as the experiments or practice carried on in 
the year 1773, on Woolwich Common ; in which all the sizes 
of mortars were used, and a variety of small charges of puw- 
der. But they were all at the elevation of 45° ; consequent- 
Jy these are defective in the higher charges, and in all the 
other angles of elevation. 

Other experimenis were also carried on in the same place 
in the years 1784 and 1786, with various angles of elevation 
indeed, but with only one size of mortar, and only one charge 
of powder, and that but a small one too; so that all those 
nearly agree with the parabolic theory. Other experiments 
have also been carried on with the ballistic pendulum, at dif- 
ferent times ; from which have been obtained some of the laws 
for the quantity of powder, the weight and velocity of the 
ball, the length of the gun, &c. Namely, that the velocity of 
the ball varies as the square root of the charge directly, and 
as the square root of the weight of ball reciprocally ; and 
that, some rounds being fired with a medium length of one- 
pounder gun, at 15° and 45° elevation, and with 2, 4, 8, 
and 12 ounces of powder, gave nearly the velocities, ranges, 


and times of flight, as they are here set down in the following 
Table. } Ae 


EEE 




















Elevation | Velocit Time of 
FoNaers. of gun. | of ball. Benge; flight. 
OZ. feet. feet. 
2 15° 660 4100 9” 
4 15 1230 5100 12 
8 15 1640 | 6000 144 
12 16; 1680 6700 15L 
2 


Ab FS 860° b10Ue 7 S14" 








106. But as we are not yet provided with a sufficient 
number and variety of experiments, on which to establish 
true rules for practical gunnery independent of the parabolic 
theory, we must at present content ourselves with the data of 

some 


142 OF MOTION, FORCES, &¢. 


fome one certain experimented range and time of flight, at a 
given angle of elevation ; and then by help of these, and the 
 tules im the parabolic theory, determine the like circumstances 
for other elevations that are not greatly different from te 
former, assisted by the following practical rules. 


SOME PRACTICAL RULES IN GUNNERY. 
1, To find the Velocity of any Shot or Shell. 


Ruxe. Divide double the weight of the charge of powder 
by the weight of the shot, both in Ibs. Extract the square 
- toot of the quotient. Multi ply that root by 1600, and the 
product will be the velocity in feet, or the number of feet the 
shot passes over per second. 

Or say As the root of the weight of the shot, is to the root 
of double the weight of the powder, so is 1600 feet, to the 
velocity. 


Il. Given the range at one Elevation; to find the Range at 
Another Elevation. 


Rure. As the sine of double the first elevation, is to its 
fange ; so is the sine of double another elevation, to its 
range. 


TH. Given the Range for One Charge; to find the Range for 
Another Charge, or the Charge for Another Range. 


Ruve. The ranges have the same proportion as the charges ; 
that is, as one range is to its charge, so is any other range to 
its charge: the elevation ef the piece_being the same in both 
cases. 

107. Example 1. If a ball of 1 Ib. acquire a velocity of 
1600 feet per second, when fired with 8 ounces of powder ; 
it is required to find with what velocity each of the several 
kinds of shells will be discharged by the full charges of pow- 
der, viz. 





Nature of the shells in inches ee 54, 42 

Their weight in Ibs. - - 196} -90| 48) 16 | 8 

Charge of powderin lbs. + 4) 4 Qi 2-1 1 
Ans. The velocities are + 7 477| 469| 566 566 








108. Exam. 2. If a shell be found to range 1000 yards, 
when discharged at an elevation of 45°; how far. will it 
range 


PRACTICAL GUNNERY. 143 


-tange when the ‘elevation is 30° 16’, the charge of powder 
being the same? _ Ans, 2612 feet, or 871 yards, 


109. Exam. 3. The range of a shell, at 45° elevation, 
being found to be 3750 feet ; at what elevation must the piece 
be set, to strike an object at the distance of 2810 feet, with 
the same charge of powder ? | : 

5 ? Ans. at 24° 16’ or at 65° 44”, 


110. Exam. 4. With what impetus, velocity, and charge 
of powder, must a 13-inch shell be fired, at an elevation of 
32° 12’, to strike an object at the distance of 3250 feet? 

Ans. impetus 1802, veloc. 340, charge 4lb, 71 oz. 


._ 111. Exam. 5. A shell being found to range 3500 feet 
when discharged at an elevation of 25° 12'; how far then 
will it range at an elevation of 36° 16 with the same charge 
of powder? Ans. 4332 feet. 


112. Exam. 6. If, with a charge of 91b. of powder, a shell 
range 4000 feet; what charge will suffice to throw it 3000 
feet, the elevation being 45° in both cases ?. 


Ans. 62lb. of powder. 


113, Exam. 7. What willbe the time of flight for any given 
range, at the elevation of 45° ? 


Ans. the time in secs. is } the sq. root of the range in feet, 


114. Exam, 8. In what time willa shell range 3250 feet, 
at an elevation of 32°? Ans. 11sec. nearly, 


115. Exam. 9. How far will a shot range on a plane which 
ascends 8° 15’; and another which descends 8° 15’; the im- 
petus being 3000 feet, and the elevation of the piece 32° 30'; 

Ans, 4244 feet on the ascent, 
and 6745 feet on the descent, 


116. Exam. 10, How much powder will throw a 13-inch 
shell 4244 feet on an inclined plane, which ascends 8° 15’, 
the elevation of the mortar being 32° 30’? 


Ans. 7:3765lb, or 7lb. 60z. 


-117. Exam, 11.-At what elevation must.a 13-inch mortar 
be pointed to range 6745 feet on a plane which descends 8° 
15’; the charge 72lb. of powder ? Ans. 32° 28", 

118. Exam. 12. In what time will-a 13-inch shell strike a 
plane which rises 8° 30’, when elevated 45°, and discharged 
with an impetus of 2304 feet ? Ans. 142 seconds, 

THE 


f 144 J 


THE DESCENT OF BODIES ON INCLINED PLANES 
AND CURVE SURFACES.—THE MOTION OF » 
_PENDULUMS. 


PROPOSITION XXIIl. 


119. If a weight w be Sustained on an Inclined Plane az by @ 
Power P, acting in a Spas wP, Parallel to the Plane, Then 
_ The Weight of the Body, w The Length as. 
The Sustaining Power p, and | The Heigth sc, and 
The Pressure on the Plane, p, | The Base ac, 
are respectively as of the Plane. 








For, draw cp perpendicular 
to the plane. Now here are 
three forces, keeping one an- 
other in equilibrio ; namely, the 
weight, or force of gravity, act- 
ing perpendicular to ac, or pa- 
rallel to Bc; the power acting A C 
parallel to ps; and the pres- 
sure perpendicular to az, or parallel to pc : but when three 
forces keep one another in equilibrio, they are proportional 
to the sides of the triangle cpp, made by lines in the direc- 
tion of those forces, by prop. 8; therefore those forces are 
to one another as BC, BD, CD. But the two triangles anc, csp, 
are equiangular, and have their like sides proportional ; there- 
fore the three Bp, Bc, cp, are to one another respectively as 
the three as, pc, ac; which therefore are as the three forces 


WwW, P, p. | 
120. Corol. 1. Hence the iralwht W, power P, and pressure 


p, are respectively as _ radius,. sine and cosine, 
of the plane’s elevation Bac above the horizon. 





For, since the sides of triangles are as the sines of their 
opposite angles, therefore the three AB, BC, Ac, 


are respectively as - - - - - sin. C, Sin. 4, Sin. B, 
eras - - - - + «= «= + = radius, sine, cosine, 


of the angle 4 of elevation. 


Or, the three forces are as Ac, cD, AD ; ht Sailaeaanae to 
their directions. 
121. Corol. 2. The power or relative weight that urges a 


body w down the inclined plane is = — x ws; or the force 
with 


DESCENTS ON INCLINED PLANES. 145 


with which it descends, or endeavours to descend, is as the 
sine of the angle a of inclination. - 


122; Roel. 3. Hence, if there be 
two planes of the same height, and two 
bodies be laid on them which are pro- 
portional to the lengths of the planes ; 
they will have an equal tendency to 
descend down the planes. And con-~ 
sequently they will mutually sustain each other if they be con- 
nected by a string acting parallel to the planes: 

123. Corol. 4. In like manner, 
when the power P acts in any 
other direction whatever, we; by 
drawing cpE perpendicular to the 
direction wr, the three forces in 
equilibrio, namely, the weight w, 
‘the power Pp, and the pressure on 
the plane, will still be respectively 
as AC, CD, AD, drawn perpendicular 
to the direction of those forces. 








PROPOSITION XXIV, 


124. If a Weight w on an Inclined Plane as, be in Equilibrio 
with another Weight wp hanging freely; then of they be set 
a-moving, their Perpendicular Velocities, in that Place, will 
be Reciprocally as those Weights. 


Let the weight w descend a very 
small space, from w to a, along the 
plane, by which the string prw will 
come into the position pra. Draw 
wi perpendicular to the horizon ac, 
and we perpendicular to ar: then 
wu will be the space perpendicularly 
descended by the weight w ; and ae, 
or the difference between Fa and rw, 
will be the space perpendicularly ascended by thi weight e% 
and their perpendicular velocities are as those spaces WH and 
ac passed over in those directions, in the same time. Draw 
) ae perpendicular to ar, and p1 perpendicular to ac. 

hen, 





in the sim. figs. acwH and AEDI, AG . WH ?: AE: DI3 
and in the sim. tri. arc, Dic, AC: CD :{ AE: DI; 
but, by cor. 4, prop 23, £0 CD crew OP 3 
therefore, by equality, AGT WHS WPS 


Vow Rie | 8 3 20 That 


146 | OF MOTION, FORCES, &c. 


That is, their perpendicular spaces, or velocities, are re- 
ciprocally as their weights or masses. 
125. Corol. 1. Hence it follows, that if any two bodies be 
in equilibrio on two inclined planes, and if they be set a- 
moving, their perpendicular velocity will be reciprocally as 
their weights. Because the perpendicular weight which sus- 
tains the one, would also sustain the other. 


126. Corol. 2. And hence also, if two bodies sustain each 
other in equilibrio, on any planes, and they be put in motion ; 
then each body multiplied by its perhenairaln velocity, will 
BIve equal products. : 


PROPOSITION XXV.__ 


127. The Vi elocity acquired by a Body descending freely down 
an Inclined Plane as, 1s-to the Velocity acquired by a Body 
falling Perpendicularly, m.the same Time ; as the Height of 
the Plane Bc, is to its Length aw. 


For the force of gravity, both per- | P 
pendicularly and on the plane, is con- D 
stant ; and these two, by corol. 2, prop. 
23, are to each other as ap tozc. But, 
by art. 28, the velocities generated by 
any constant forces, in the same time, Pa Ue C 
are as those forces. Therefore the velocity down BA Is to 
the velocity down sc, in the same time, as the force on Ba to 
the force on sc: that is, as Bc to Ba. 


128. Corol. 1. Hence, as the motion down an inclined 
plane is produced by a constant force, it will be a motion 
uniformly accelerated ; and therefore the laws before laid 
down for accelerated motiens in general, hold good for mo- 
tions on inclined planes ; such, for instance, as the following : 
That the velocities are as the times of descending from rest ; 
that the spaces descended are as the squares of the velocities, 
or squares of the times ; and that if a body be thrown up an 
inclined plane, with the velocity it acquired in descending, it 
will lose all its motion, and ascend to the same height, in the 
game time, and will repass any point of the plane with hs 
same velocity as it passed it in RESC RRR. 


129. Corol. 2. Hence also, the space descended down an * 
_ inclined plane, is to the space descended perpendicularly, 1 in 
the same time, as the height of the plane es, to its length 
AB, OF as the sine of inclination to radius. For the spaces 

: described 


4 


DESCENTS ON INCLINED PLANES. 147 


described by any forces, 1 in the same hone are as the forces, 
or as the velocities. 

130. Corol. 3. Consequently the yalbetite and spaces de- 
scended by bodies down different inclined planes, are as the 
sines of elevation of the planes. 


131. Corol. 4. If cp be drawn perpendicular te as ; then 
while a body falls freely through the perpendicular space sc, 
another body will in the same time, descend down the part of 
the plane sp. For bysimilartriangles, - . - ° 
BC : BD : : BA: BC, that is, as the space descended, by corol. 2. 

Or, im any right-angled triangle Bpc, ARB 
- having its hypothenuse Bc perpendicular 
to the horizon, a body will descend down 
any of its three sides Bp, sc, pc, in the 
same time. And therefore, if on the di- E 
ameter gc a circle be described, the time 
of descending down any chords kp, Br, 

BF, Dc, Ec, Fc, &c. will be ali equal, and 
_ each equal to the time of falling freely 
through the perpendicular diameter sc. 






PROPOSITION XXVI. 


\ 


132. The Time of descending down the inclined Plane a, is to 
the Time of falling through the Height of the Plane zc, as the 
Length pa is to the Height BC. 


Draw cp perpendicular to AB. 
Then the times of describing Bp and 
Bc are equal by the last corol. Call 
that time ¢, and the time of ‘describ- 
ing Ba call 7. 

Now, because the space describ- A Cc 
ed by constant forces, are as the squares of the times ; there- 
fore t?.: 7? :: BD: BA. 

But the three sp, Bc, BA, are in continual: proportion : 

_ therefore, pp : BA : : BC? : : BA? ; 
sents by equality. 2: TA2 5 Be? : Bal, 
; 2 te J ag 3 BA. 


133. Corol. Hence the times of descending down different . 
plantas of the same height, are to one another as the lengths 
of the planes. — 


PROPOSITION 


148 | OF MOTION, FORCES, &c. 


; PROPOSITION XXVIL 


134. A Body acquires the Same Velocity in descending down any 
Inclined Plane wa, as by falling see eee’ Higowahs the 
Height of the Plane sc. 


For, the velocities generated by any constant forces, are 
in the compound ratio of the forces and times of acting. 
But if we put 

F to denote the whole force of gravity In BC, 

f the force on the plane az, } 2 

t the time of describing bc, and | D 


T the time of descending down AB ; 
then by art. 119, F if : BA ¢ BC 5 
and by art. 132, ¢, 23 BO 2 BA c 


theref. by comp. rt fe od sds me 
That is the compound ratio of the forces and times, or the 
ratio of the velocities, is a ratio of equality. 


a ow 





135. Corol. 1. Hence the velocities acquired, by bodies 
descending down any planes, from the same height, to the 
same horizontal line are equal. 


~ 136. Corol. 2. If the velocities be equal, at any two equal 
altitudes, p, x; they will be eqial 8 at all other equal aii 
Ky) Goi 


137. Corol. 3. Hence also the velocities veaaived, by de- 
scending down any planes, are as the square roots of the 
heights, 


PROPOSITION XXVIII. 


138. Ifa Body descend down any Number of Contiguous Planes, 

AB, BC, cD; wt will at last acquire the Same Velocity, as a 
Body folling perpendicularly through the Same Height xn, 
‘supposing the Velocia) not Buckets by changing from one Plane 
to another. 


_ Provuce the planes p Dc, cB, to a AMG 
meet the horizontal line Ea pro- , 
uced in F and c. Then, by 
art. 135, the velocity at z is the 
same whether the body descend 
through az orFe. And therefore 
the velocity atc willbethe same, 
aes the body descend through 4 apc or through FC, 
which 





DESCENTS ON INCLINED PLANES. 149° 


which is also again, by art. 135, the same as by descending 
through ec. Consequently it will have the same velocity at p, 
by descending through the planes as, Bc, cp, as by descending 
through the plane cp; supposing no obstruction to the mo- 
tion by the body impinging on the planes at 8 and c: and this 
again, is the same velocity as by descending through the same 
perpendicular height ep. 


139. Corol. 1. Ifthe lines ascp, &c. be supposed indefinite- 
ly small, they will form a curve line, which will be the path 
of the bedy ; from which it appears that a body acquires alse 
the same velocity in descending along any curve, as in falling 
perpendicularly through the same height. 


140. Corol. 2. Hence also, bodies acquire the same velo- 
city by descending from the same height, whether they de- 
scend perpendicularly, or down any planes. or down any curve 
orcurves. And if their velocities be equal, at any one height, 
they will be equal at all other equal heights. Therefore the 
velocity acquired by descending down any lines or curves, are 
as the square roots of the perpendicular heights. 


141. Corol. 3. Anda body, after its descent through any 
curve, will acquire a velocity which will carry it to the same 
height through an equal curve, or through any other curve 
either by running up the smooth concave side, or by being 
retained in the curve by a string, and vibrating like a pendu- 
lum: Also, the velocities will be equal, at all equal altitudes; 
and the ascent and descent will be performed in the same time, 
if the curves be the same. 


PROPOSITION XXIX. 
142. The Times in which Bodies descend through Similar Parts 
of Similar Curves, anc, abe, placed alike, are as the Square 


Roots of their Lengths. 


Tuaris, the time in ac is to the time in ac, as ,/ Ac to,/ac. 
For, as the curves are similar, they may - 


be considered as made up of an equal D @ A 
number of corresponding parts, which ; 
are every where, each to each, propor- Cc 


tional to the whole. And as they are plac- 
ed alike, the corresponding small similar 
parts will also be parallel to each other. 
But the time of describing each of these pairs of correspond- 
ing parallel parts, by art, 128, are as the square roots of their 
oi : lengths, 


150 | OF MOTION, FORCES, 


lengths, which by the supposition, are as 4/ ac to 4/ ac, the 
roots of the whole curves. Therefore, the whole thes, are 
in me same ratio of ,/ acto 4/ ac. 

. Corol. 1. Because the axes pc, no, of stiles curves, 
at as * the lengths of the similar parts ac, ac; therefore the 
times of descent in the curves ac, ac, are as 4/ vc to v DC, 
or the square roots of their axes. 

144. Corol. 2. As itis the same thing, whether the bodies 
run down the smooth concave side of the curves, or be made 
to describe those curves by vibrating like a pendulum, the 
lengths being pc, pc; therefore the times of the vibration of 
pendulums, in similar arcs of any curves are as the square 
- roots of the lengths of the pendulums. 


SCHOLIUM. 


345. Having in the last corollary, mentioned the pendulum, 

{may not be improper here to add some remarks concern- 
ing it. 

A pendulum consists of a ball, or any 
other heavy body 8, hung bya fine string A. 
or thread, ‘oVerble: about .a centre 4, 
and describing the arc cpp; by which 
vibration the same motions happen to this 
heavy body, as would happen to any 
body descending by its gravity along the 
spherical superficies cep, if that superfi- 3 
cies were perfectly hard and smooth. {f the pendulum be 
carried to the situation ac, and then let fall, the ball in de- 
scending will describe the arc cp; and in the point B it 
will have that velocity which is’ acquired by descending 
through cs, or by a body falling freely through es. This 
velocity will be sufficient to cause the ball to ascend through 
an equal arc gp, to the same height p from whence it fell 
atc; having there lost all its motion, it will again begin to 
descend by its own gravity ; and in the lewest point 8 it will 
acquire the same velocity as before ; which will cause it to 
_re-ascend to c; and thus, by ascending and descending, it will 
perform continual vibrations on the circumference cpp. And 
if the motions of pendulums met with no resistance from 
the air, and if there were no friction at the centre of mo- 
- tion a, the vibrations of pendulums* would never cease. 
’ But from these obstructions, though small, it happens, that 
the velocity of the ball in the. point B is a little diminished 
in every vibration; and consequently it does not return 


precisely to the same points c or p; but the arcs described con- 
tinually 





' PENDULUMS. © 151 


tinually become shorter and shorter, till at length they are 

insensible ; unless the motion be assisted by a mechanical 

contrivance, as in clocks, called a maintaining pone 
DEFINITION. 


146. If thie'c cir- 
cumference of a 
circle be rolled on 
a right line begin- 
ning at any point | 
a, and continued 
till the same point 
a arrive at the line. 
again, making just one ESN ed and thereby measuring out 
a straight line apa equal to the circumference of the circle, 
while the point a in the circumference traces out a curve line 
acaGa: then this curve is called a cycloid ; and some of its 
properties are contained in the following lemma. 

LEMMA. 

147. If the generating or revolving circle be placed in the 
middle of the cycloid, its diameter coinciding with the axis 
4B, and from any point there be drawn the tangent'cr, the or- 
dinate cpe perp. to the axis, and the chord of the circle ap ; 
Then the chief properties are these : 

The right line cp= the circular arc ap ; 

Lhe cycloidal arc ac= double the chord ap; 

The semi- cycloid aca= double the diameter ap, and 

The tangent cr is parallel to the chord ap. | 





x 


PROPOSITION XXX. 


148. When a Pendulum vibrates in a cyctoid; the Time of one ~ 
Vibration, ts to the Time in which a Body falls through half 
the Length of the Penduluin, as the Circumference of a Cire 
cle is to its Diameter. 


Let asa be the cycloid ; 
‘ps its axis, or the diameter mG 
of the generatingsemicircle 
DEB ; cB = 2pB the length 
of the pendulum, or radius 
of curvature ats. Let the 
ball descend from Fr, and . 
in vibrating, describe the | 
arc Ff. Divide Fs into in- 
numerable small parts, one 
of which is eg ; draw Fer, 
GM, git, perpendicular to 





DE, 


152 OF MOTION, FORCES, &c. 


ps. On wp describe the se- . C 
-micircle tmp, whose cen- | 

tre is o; draw mp: parallel 
to pg; also draw the chords | 
BE, BH, EH, and the radius 
om. 


_BHK, are equiangular; there- 
fore BK : BH: : BH : BE, Or 
BH? = BK . BE, OF BH = 4/ 
(BK. BE). 

And the EqUSAEHIAr triangles mmp, MON, give 
Mp: Mm::Mn:Mo. Also, by the nature of the avelold. Hh 
is equal to cg. 

If another body descend down the chord zs, it will have 
the same velocity as the ball in the cycloid has at the same 
height. So that xk and eg are passed over with the same ve- 
locity, and Coca neDny the time in passing them will be as 
their lengths eg, Kk, or.as nh to Kk, or BH, to BK by similar 
triangles, or J (BK . BE) to BK, Or 4/ BE to ¥ BK, OF aS 4/BL 
to / pn by similar triangles. 

‘That is, the time in cg: time in kk: : ¢/ BL: 4/ BN. 

Again, the time of describing any space with a uniform mo- 
tion, is directly as the space, and reciprocally as the velocity ; 
also, the velocity in x or kk, is to the velocity at.B, as / EK 
to ,/ EB, or aS 4/ LN to,/ 1B; and the uniform velocity for 
EB Is equal to half that at the point 8, therefore the time in Kk: 

BRB) ot ONE Ole 
VIN EA/LBo > 4/LN 2A/LB 

by sim. tri): : Nn or MP: 2,/ (BL . LN.) 

That is, the time in Kk: time in eB: : Mp : 2,/ (BL... EN.) 
But it was, time in cg : time in kk: : ,/ BL :,/ BN; theref. 
by comp. time in eg: time in EB: : MP: 2,/(BN. NL) or 2nM. 
~- But, by sim. tri. mm : 2om or BL :: MP: 2NM. 

Theref. time in cc: time in EB: : Mm: BL. 

Consequently the sum of ail the times in all the cg’s, is to 
_ the time in eB, or the time in pg, which ts the same thing, as 
the sum of all ‘the mm’ S, is to LB; 4 
that is, the time in rg + time in DB $8 EMU sLB, 
and the timein -_ FB: time in DB: :. LMB : LR, 

or the time in ref : time in DB: : 2LMB: LB. 

. That is, the time of one whole vibration, 

is to the time of falling through half cx, 
as the circumference of any circle, 
is to its diameter. 








time in EB:: 


149. Corol. 


-. PENDULUMS, 153 


149. Corel, 1. Hence all the vibrations of a pendulum in 
a cycloid, whether great or small, are performed in the same 
time, which time is to the time of falling through the axis, or 
half the length of the pendulum, as 3:1416, to 1, the ratio of 
the circumference to its diameter ; and hence, that time is easily 
foundthus. Put p = 3-1416, and | the length of the pendu; 
lum, also g the space fallen by a heavy body in 1” of time. 


then 4/ g : /}l:: i vz the time of falling through 11, 


theref. 1: p : ein ; IVs which therefore is the time of 


one vibration of the ees 

150. Andif the pendulffm yibrate in a small arc of a eircles ; 
because that small arc nearly coincides with the small cycloi- 
dal arc at the vertex s; therefore the time of vibration in 
the small are of a circle, is nearly equal to the time of vibra- 
tion in the cycloidal arc ; consequently the time of vibration 


l ay , 
in a small circular arc is equal to p4/ 3g’ where | is the Fading 


of the circle. 

151. So that, if one of these, g or J, be found is experi- 
ment, this’ Pidarers will give the other. Thus, if g, or the 
space fallen through by a heavy body in 1” of time, be found, 

then this theorem will give the length of the second pendu- 
lum. Or, if the length of the second pendulum be observed 
by experiment, which is the easier way, this theorem will give 
g the descent of gravity in 1”. Now, in the Jatitude of 
London, the length of a pendulum which vibrates seconds, 
has been found to be 392 inches; and this being written for / 


, gah: fe 94 
in theorem, it gives p ny ae = 1’: hence is found g = ip? 


=1p? X 391=193:07 inches = 16, feet, for the descent of 
gravity in 1"; which it has also been found to be, very nearly, 
by many accurate experiments, 


SCHOLIUM, 


152. Hence is found the length of a pendulum that shall 
make any number of vibrations in a given time. Or, the 
number of vibrations that shall be made by a pendulum of 
a given length. Thus, suppose it were required to find the 
length of a half-seconds pendulum, or a quarter- -seconds pen- 
dulum; that is, a pendulum, to vibrate twice in a second, or 
4 times in a second. Then since the time of vibratign is as 
the square erie gt the length, 
therefore 1 : A f39F ca/l 

Vor, I. ai op 


154 MECHANICAL POWERS. 


a as che BOF : aie 93 inches ee the length 


of the Sue deabnts pendulum. Again 1: jj. :: 391 : 24 in- 
ches, the length of the quarter-seconds ae und | 

_ Again, if it were required to find how many vibrations a 
pendulum of 80 inches long 7 Sr in atminute. Here 


4/80 : 4/39} :: 60" or I’ : 60/2 =713/313=41- 95987, 


or almost 42 vibrations in a minute. 

153. In these propositions, the thread is supposed to be 
very fine, or of no sensible weight, and the ball very small, 
or all the matter united in one point; also, the length of the 
pendulum, is the distance from the point of suspension, or 
centre of motion, to this point, or centre of the small ball. 
But if the ball be large, or the string very thick, or the vi- 
brating body be of any other figure ; then the length of the 
pendulum is different, and is measured, from the centre of 
motion, not to the centre of magnitude of the body, but to 
such a point. as that if allthe matter of the pendulum were 
collected into it, it would then vibrate.in the same time as the 
compound pendulum; and this point is called the Centre of 
Oscillation ; a point which will be treated of in what follows. 


THE MECHANICAL POWERS, &c. 


154. WEIGHT: and Power, when opposed to each other, : 

signify the body to be moved, and the body that moves it ; 
or the patient and agent. The power is the agent, which 
moves, or endeavours to move, the patient,or weight. 
155. Equilibrium, is an equality of action or force, be- 
tweén two or more powers or weights, acting against ‘each 
other, by which they destroy each other’s effects, and remain 
at rest. 

156. Machine, or Engine, is any Mechanical instrument 
contrived to move bodies. And it is Prepoved of the me- 
chanical powers. 

157. Mechanical Powers, are certain Sidple instruments, 
commonly employed for raising greater weights, or overcom- 
ing greater resistances, than could be effected by the natural 
strength without them. These are usually accounted six in 
number, viz. the Lever, the Wheel and Axle, the Pulley, the 
Inclined Plane, the Wedge, and the Screw. 

158. Mechanics, 


THE BEVER. 155 


158. Mechanies, is the science of forces, and the effects 
they produce, when applied to machines, in the motion of 
bodies. 

159. Statics, is the science of weights, eapecially when con- 
sidered in a state of equilibrium. _ 

160. Centre of Motion, is the fixed point about which a 
body moves. And the Axis of Motion, is the fixed line about 
which it moves. 

161. Centre of Gravity, is acertain point, on which a body 
being freely suspended, it will rest in any position. 


OF THE LEVER. 


162, A Lever is any inflexible rod, bar, or beam, which 
serves to raise weights, while it is supported at a point by a 
‘fulcrum or prop, which is the centre of motion. ‘The lever 
is supposed to be void of gravity or weight, to render the de- 
monstrations easier and simpler. There are three kinds of 
levers. 


163. A Lever of the First Aloe eaux mae mir SF cients Pp 
kind has the prop. c be- | Vedat 
tween the weight w and 4' | \ 
the power p. And of this 
kiad are balances, scales, 


crows, hand-spikes, scis- 
‘sors, pinchers, &c. 





164. A Lever of the Se- 
cond kind has the weight 
between the power and the 
prop. Such as oars, rud- 
ders, cutting knives fhat are 
fixed at one end, &c. 


165, A Lever of the 
Third kind has the power 
between the weight and | 
the prop. Such astongs, | x 

the bones and muscles of B66 ENUO TE. 
animals, a man rearing a PP)  # 8 | C 
ladder, &c. 





166. A 


16 MECHANICAL POWERS. 
166. A Fourth kind is some- 





times added, called the Bended © WwW Ge p 
~Lever. Asahammer drawi ing o- mais / 
a nail. ath 


167. In all these instruments the power may be represent- 
éd by a weight; which is its most tiatural measure, acting 
downward : but having its direction changed, when necessary; 
by means of a fixed pulley. : 


PROPOSITION XEXI. 


168. When the Weight and Power keép the Lever in Equilibrio, 
they are to each other Reciprocally as the Distances ef their 

Lines of Direction from the Prop... That is, rp: Ww :: cp ¢ 
cE; where cp and cE are perpendicular to wo and Ao, the 
Directions of the two Weights, or the Weight and Power vw 
and a. 


-For, draw cr parallel to ao, and 
cp parallel to wo: Also join co, 
which will be the direction of the 
pressure.on the prop c; for there 
cannot be an equilibrium unless the 
directions of the three forces all 
meet in, or tend to, the same point, 
as 0. Then, ecanse these three 
forces keep each other in equilibrio, 
they are proportional to the sides of 
the triangle cso or cro, drawn in the 
direction of those forces ; therefore | 

- P iwi: cr Fo or cB: 
But, bavduce af the parallels, “the | ; 
two triangles cpr, CEB are equiangu- 
lar, therefore - - + - + - HD SOR & 2 GR 2G: 
Hence, by equality, - - -- PiW :: Cp :-¢E. 

That is each force is reciprocally proportional to the dis- 
tance of its direction from the fulcrum. 

And it will be found that this demonstration will serve for 
all the other kinds of levers, by drawing the lines as directed. 
169. Corol. 1. When the angle a ue the angle w, then 
i$ CD :CE:::cw:ca:ip:w: Or when the two forces: act 
perpendicularly. on he lever, as two weights, &c. ; then, in 
case of an equilibrium, pv coincides with w, and & with P45 
consequently then the above proportional becomes also P.: W 
: + cw: Ga, or the distances of the two forces from the ful- 


crum, taken on the lever, are reciprocally proportional to 
those forces. 





170. Corol: 


THE LEVER: 157 


170. Corol. 2. If any force ¥ be applied to a lever at a 3 
its effect on the lever, to turn it about the centre of motion 
c, is as the length of the lever ca, and the sine of the angle 
of ao. For the perp. ce is as'caXsAa. 

171, Corol. 3. Because the product of the extremes is 
equal to the product of the means, therefore the product of 
the power by the distance of its direction, is equal to the pro- 
duct of the weight by the distance of its direction. 

That is, PX ce=w Xop. . 

172. Corol. 4. If the lever, with the weight and power 
fixed to it, be made to move about the centre c ; the momen- 
tum of the power will be equal to the momentum of the 
weight ; and their velocities will be in reciprocal proportion 
to each other. For the weight and power will describe cir- 
cles whose radii are the distances cp, ce; and since the cir- 
cumferences or spaces described, are as the radii, and also 
as the velocities, therefore the velocities are as the radii cp, 
- ce; and the mementa, which are as the masses and velocities, 
-are as the masses and radii; that is, as rXce and eee 
which aré equal by cor. 3. 

173. Corol. 5. Ina straight lever, kept in equilibrio by 
a weight and power acting perpendicularly ; then, of these 
three, the power, weight, and pressure on the prop; any one 
is as the distance of the other two. 

174; <Curol. 6... If A B GC ass iG 
several weights, P,Q, RB, mR | sane Racecar eae or 
s, act on a straight le- ' 
ver, and keep it in equi- 
librio ; then the sum of | ~ " | 
the products ononeside ~P 0. RK s 
of the prop. will be 
equal to the sum on the 
other side, made by multiplying each weight by its distance ; 
hamely, 

PXac+e Xpo=r Xvc-+s XEC: 

For, the effect of each weight to turn the lever, is ‘as the 
weight multiplied by its distance ; and in the case of an equi- 
librium, the sums of the effects, or of the produces on both 
sides are équal. 

175. Corol. 7. Because, when ra D 


two weights q and R are in equi- BL au? 
librio, a: R:: cp: cB; therefore, | | 
© > 


0. ts 





by Sibimritin EON atria: : CD, 
and, atR:Ri:BD: 


158 MECHANICS, 


That is, the sum of the weights is to either of them, as the 
sum of their distances is to the distance of the other. 


~ SCHOLIUM. 


176. On the foregoing prin- 
ciples depends the nature of 
scales and beams, for weigh- 
ing all sorts of goods. For, 
if the weights be equal, then 
will the distances be equal al- 
80, which gives the construc- 
tion of the common scales, 
which ought to have these pro- 
perties : = = 

Ist. That the points of suspension of the scales and the 
centre of motion of the beam, 4,8, c, should be in a straight 
line: 2d, That the arms as, xc, be of an equal length : 3d, 
That the centre of gravity be in the centre of motion B, or a 
little below it: 4th, That they be in equilibrio when empty : 
‘5th, That there be as little friction as possible at the centre B. 
A defect in any of these properties, makes the scales either 
imperfect or false. But it often happens that the one side of 
the beam is made shorter than the other, and the defect cover- 
ed by making that scale the heavier, by which means the 
scales hang in equilibrio when empty; but when they are 
charged with any weights, so as to be still in equilibrio, those 
weights are not equal; but the deceit will be detected by 
changing the weights to the contrary sides, for then the equi- 
librium will be immediately destroyed. | 

177. To find the true weight of any body by suph: a false 
balance :—First weigh the body i in one scale, and afterwards 
weigh it in the other; then the mean proportional between 
these two weights, will be the true weight required. For, if 
any body 6 weigh w pounds or ounces in the scale p, and only 
e pounds or ounces in the scale x: then we have these two 
equations, namely, ap. b=rc. w. 

and pc. b=aB ..w; 
the product of the two is aB..Bc . b°?=AaB. BC. Ww; 
hence then - - - be whe, 7 
and - - - b2=,/ww, 
the mean proportional, which is the trae weight of the body 0. 





_ 178. The Roman Statera, or Steelyard, is also a lever, but 
of unequal brachia or arms, so contrived, that one weight 


only may serve to weigh a great many, by sliding it back- 
ward 


“ag 
y 


THE WHEEL AND AXLE. 159 


ward and forward, to different distances, on the longer arm of 
the lever; and it is thus scciibiuiaibess Rice 





Let as be the steelyard, and c its centre of motion, whence 
the divisions must commence if the two arms just balance 
each other: if not, slide the constant moveable weight 1 
along from gs towards c, till it just balance the other end 
without a weight, and there make a notch in the beam, 
marking it with acipher 0. Then hang on at aa weight w 
equal to 1, and slide 1 back towards B till they balance each 
other; there notch the beam, and mark it with 1. Then 
make the weight w double of 1, and sliding 1 back to balance 
it, there mark it with 2. Do the same at 3, 4, 5, &c. by 
making w equal to 3, 4, 5, &c. times1; and the beam is 
finished.. Then to find the weight of any body 6 by the 
steelyard ; take off the weight w, and hang on the body 6 
at a: then slide the weight 1 backward and forward till it just 
balance the body 6, which suppose to be at the number 5; 
then is 5 equal to 5 times the weight of 1. So, if 1 be one 
pound, then 6 is 5 pounds ; but if 1 be 2 pounds, then 6 is 10 
pounds ; and so on. 


—=—— 


~ . . OF THE WHEEL AND AXLE. 
' PROPOSITION XXXU. 


179. In the Wheel-and-Azle ; the Weight and Power will be 
in Equilibrio, when the Power p is to the Weight w, Reci- 
procally as the Radi of the Circles where they act; that is, 
as the Radius of the Axle ca, where the Weight hangs, to 
the Radius of the Wheel cp, where the Power acts. That is, 
Pt W 2% Ass GBs 
HBRE the cord, by which the power P acts, goes aloe 

the 


160 _ MECHANICS, 


the circumference of the wheel, while 
that of the weight w goes round its 

_ axle, or another smaller wheel, attach- 
ed to the larger, and haying the same 
axisorcentrec. Sothat Ba isa lever 
moveable about the point c, the power — 
P acting always at the distance gc, and 
the weight w at the distance ca; there- 
fore p: wet cA: cB. 

180. Corol. 1. If the wheel be put 
in motion ; then, the spaces moved 
being as the circumferences, or as the radii, the velocity of 
w will be to the velocity of P, as. ca to cB; that is, the 
weight is moved as much slower, as it is heayier than the 
power; so that what is gained in power, is lost in time. 
And this is the universal property of all machines and en- 
gines. 

181. Corol. 2. If the power do not act at right angles 
to the radius cs, but obliquely ; draw cp perpendicular to 
the direction of the power; then, by the nature of the lever, 
Pos w sos 2 on, : 





* 


SCHOLIUM, — 


182. To this power be- 
long all turning or wheel 
machines, of different radii, 
Thus, in the roller turning | 
on the axis or spindle cr, 
by the handle cpp; the 
power applied at B is to 
the weight w on the roller 
as the radius of the roller 
is to the radius cp of the 
handle. 

183. And the same for all cranes, capstans, windlasses, and 
such like; the power being to the weight, always as the ra- 
dius or lever at which the weight acts, to that at which the 
power acts ; so that they are always in the reciprocal ratio of 
their velocities. And to the same principle may be referred 
the gimblet and auger for boring holes. 

184. Butall this, however, is on supposition that the ropes 
or cords, sustaining the weights, are of no sensible thickness, 
For, if the thickness be considerable, or if there be several 
folds of them, over one another, onthe reller or barrel; then 
ye must measure to the middle of the outermost rope, a 

: 16 





THE WHEEL AND AXLE. 161 


the radius ef the roller; or, to the radius of the roller we 
must add half the thickness of the chord, when there is but 
one fold. 

185. The wheel-and-axle has.a great advantage over the 
simple lever, in point of convenience. For a weight can be 
raised but a little way by the lever; whereas, by the continual 
turning of the wheel and roller, the weight may be raised to 
any height, or from any depth. i 

186. By increasing the number of wheels too, the power 
may be multiplied to any extent, making always the less 
wheels to turn greater ones, as far as we please ; and this is 
commonly called Tooth and Pinion Work, the teeth of one 
circumference working in the rounds or pinions of another, 
to turn the wheel. And then, in case of an equilibrium, the 
power is to the weight, as the continual product of the radii 
of all the axles, to that of all the wheels. So, if the power P 





turn the wheel a, and this turn the small wheel or axle r, 
and this turn the wheel s, and this turn the axle 1, and this 
turn the wheel v, and this turn the axle x, which raises the 
weight w; then P: w::.cB.DE.FG: AC.BD.EF. And in 

the same proportion is the velocity of w slower than that of rp. — 
Thus, if each wheel be to its axle, as 10 to 1; thene: w:: 
13: 103 or as 1 to 1000. So that a power of one pound will 
balance a weight of 1000 pounds ; but then, when put in mo- 
tion, the power will move 1000 times faster than the weight. - 


Vor. He. | Ly BOs. | OF 


162 MECHANICS. 
OF THE PULLEY. 


187. A Puutey is a small ee commonly made of wood 
or brass, which turns about an iron axis passing through the » 
centre, and fixed in a block, by means of a cord passed round 
its circumference, which serves to draw up any weight. The 
pulley is either single, or combined together, to increase the 
power. It is also either fixed or moveable, according as it 
is fixed to one place, or moves up and down with the Reigit 
and power 


PROPOSITION XXXIifI. 


188. if a Power sustain a Weight by means of a Fixed Pulley; 
| the Power and Weight are Equal. 


For, through the centre c of the pulley 
draw the horizontal diameter ap: then will 
aB represent a lever of the first kind, its 
prop being the fixed centre c; from which 
the points a and s, where the power and — 
weight act, being equally distant, the pow- 
er P is ‘consequently equal to the weight 


489. Corol. Hence, if the pulley be put 
in motion, the power P will descend as fast 
as the weight w ascends. So that the 
‘power is not increased by the use of the 
fixed pulley, even though the rope go over 
several of them. It is, however, of great service in the 
raising of weights, both by changing the direction of the force, 
for the convenience of acting, and by enabling a person to 
raise a weight to any height without moving from his place, 
and also by permitting a great many persons at once to exert 
their force on the rope at Pp, which they could not do to the 

~ weight itself ; as is evident in raising the hammer or weight 
oe oF a pile- -driver, as well as on many ofper occasions. 





PROPOSITION HXSLV' 


190. ie a Power sustain a Weight by means of One Moveable 
7 Pulley ; the Power ts but Half the Weight. oie 


For, here ag may be considered as a lever of the second 
kind, 


THE INCLINED PLANE. 163 


kind, the power acting ata, | 
the weight atc, andthe prop <& 
or fixed point atp; and be- “\° 
cause Pp: W:: CB: AB, and 
cp=las, therefore p=}w, 
or w=2P. 

191. Corol. 1. Hence it 
is evident, that when the 
pulley is put in motion, the 
velocity of the power will 
be double the velocity of lic , ) 
the weight, as the point Pp SL SNe 
moves twice as fast as the point c and weight w rises. _ It is 
also evident, that the fixed pulley r makes no difference in the 
power p, but is only used to change the direction ef it, from 
upwards to downwards. | 

192. Corol. 2. Hence we may estimate the effect of acom- . 
bination of any number of fixed and moveable pulleys ; by 
which we shall find that every cord going over a moveable 
pulley always adds 2 to the powers ; since each moveable pul- 
ley’s rope bears an equal share of the weight ; while each rope 
that is fixed to a pulley, only increases the power by unity. 











Here p=iw. 


: OF THE INCLINED PLANE. 

193. THE Incuinep Puane, is a plane inclined to the 
horizon, or making an angle with it. _ It is often reckoned one 
of the simple mechanic powers ; and the double inclined plane 
makes the wedge. It is employed to advantage in raising 
heavy bodies in certain situations, diminishing the powers that 

ustain them by laying them on the inclined planes. 
PROPOSITION 


164 | MECHANICS. 
PROPOSITION XXXV. 


194. The Power gained by the Inclined Plane, is in Proportion 
as the Length of the Plane is to its Height. That is, when a. 
Weight w vs sustained on an Inclined Plane; sc, by a Power 
P acting wn the Direction pw, parallel to the Plane ; then the 
Weight w, ws in proportion to the Power p, as the Length of 
the Plane 1s to rts Height; that is, w:P::8c: AB. 


For, draw az perp. to 
the plane sc, or to pw. 
Then we are to consider 
that the body w is sustain- 
ed by three forces, viz. 
Ist, its own weight or the 
force of gravity, acting perp. to ac, or parallel to pa ; 2d, by 
the power P, acting in the direction wn, parallel to ec, or BE ; 
and 3dly, by the re-action of the plane, perp. to its face, or 
parallel to the line ra. But when a body is kept in eqnili- 
brio by the action of three forces, it has been proved, that 
the intensities of these forces are proportional to the sides 
of the triangle aBe made by lines drawn in the directions of 
their actions ; therefore those forces are to one another as 





the three lines - - - - AB, BE, AE; that is, 
the weight of the body w is as the line azn, 
the power P Is as the line . - BE, 


and the pressure on the plane as the line ar. te 
But the two triangles ape, aBc are equiangular, and have 
therefore their like sides proportional; that is, 


the three lines - - - - AB, BE, AE, 
are to each other respectively as the three Bc, aB, Ac, 
or also as the three - ‘= - BC, AE, CE, 


which therefore are as the three forces w, PF, p, 
where p denotes the pressure on the plane. Thatis,w:P:: 
BC : AB, or the weight is to the power, as the length of the 
plane is to its height. : 

See more on the Inclined Plane, at p. 144, &c. 


195. Scholium. The Inclined plane comes into use in some 

situations in which the other mechanical powers cannot be; 
conveniently applied, or in combination with them. As, in 

sliding heavy weights either up or down a plank or other plane 

laid sloping : or letting large casks down into a cellar, or draw- 

~ Ing them out of it. Also, in removing earth from a lower 

situation to a higher by means of wheel-barrows, or other- 

wise, as in making fortifications, &c. ; inclined planes, made 

of boards, laid aslope, serve for the barrows to run upon. 


THE WEDGE. 165 


Of all the various directions of drawing bodies up an inclined 
plane, or sustaining them on it, the most favourable is where 
it is parallel to the plane*sc, and passing through the centre of 
the weight ; a direction which is easily given to it, by fixing . 
a pulley at p, so that a cord passing over it, and fixed to the 
weight, may act or draw parallel to the plane. In every other 
position, it would require a greater power to support the body 
on the plane, or to drawit up. For if one end of the line be 
fixed at w, and the other end inclined down towards zg, below 
the direction wp, the body would be drawn down against the 
plane, and the power must be increased in proportion to the 
greater difficulty of the traction. And, on the other hand, if 
the line were carried above the direction of the plane, the 
power must be also increased ; but here only in proportion as 
it endeavours to lift the body off the plane. 

If the length sc of the plane be equal to any number of 
times its perp. height an, as suppose 3 times ; then a power 
p of 1 pound hanging freely, will balance a weight w of 3 
pounds, laid on the plane: and a power P of 2 pounds, will 
balance a weight w of 6 pounds ; and so on, always 3 times 
asmuch. But then if they be set a-moving, the perp. descent 
of the power P, will be equal to 3 times as much as the perp. 
ascent of the weight w. For, though the weight w ascends 
up the direction of the oblique plane, se, just as fast as the 
power p descends perpendicularly, yet the weight rises only 
the perp. height as, while.it ascends up the whole length of 
the plane gc, which is 3 times as much ; that is, for every foot 
of the perp. rise, of the weight, it ascends 3 feet up in the 
pera of the plane, and the power rv descends as much, or 3 

eet. 


ie 


OF THE WEDGE. 


196. THE Wence is a piece 0 
wood or metal, in form of half a rec- 
tangular prism. a¥r or Ba is the 
breadth of its back; ce its height ; 
ec, Bc its sides: and its end csc is 
composed of two equal inclined 
planes ccr, Bce. 





ce ‘ 
PROPOSITION 


166 MECHANICS. 
PROPOSITION XXXVI. 


197. When a Wedge is in Equilibrio ; the Power acting against 
the Back, is to the Force acting Perpendicularly against either 
Side, as the Breadth of the Back an, is to the Length of the 


Side ac or BC. 


For, any three forces, which sustain one 
another in equilibrio are as the correspond- 
ing sides of a triangle drawn perpendicular Hy! 
to the directions in which they act. But .qqyiill 
ap is perp. to the force acting on the back, \\\\Wi 
to urge the wedge forward ; and the sides \\ 
ac, Bc are perp. to the forces acting on 
them ; therefore the three forces are as aB, 
AC, BC? fet pa 

198. Corol. The force on the back, ¢ as, 

Its effect in direct. perp. to ac, ) ac, 
And its effect parallel to as ; BC, ) 

.. are as the three lines / whichare per. to them. 

And therefore the thinner a wedge is, the greater is its ef- 
fect in splitting any body, or in overcoming any resistance 
against the sides of the wedge. 





SCHOLIUM. 


199. But it must be observed, that the resistance, or the 
forces. above-mentioned, respect one side of the wedge only. 
For if those against both sides be taken in, then, in the fere- 
going proportions, we must take only half the back ap, or else 
we must take double the line ac or pe. 

In the wedge, the friction against the sides is very great, at 
least equal to the force to be overcome, because the wedge 
retains any position to which it is driven; and therefore the 
resistance is double by the friction. But then the wedge has 
a great advantage over all the other powers, arising from the 
force of percussion or blow with which the back is struck, 
which is a force incomparably greater than any dead weight or 
pressure, such as is employed in other machines. And accord- 
ingly we find it produces effects vastly superior to those of 
any other power ; such as the splitting and raising the largest 
and hardest rocks, the raising and lifting the largest ship, by 
driving a wedge below it, which a man can do by the blow of a 
mallet: and thus it appears that the small blow of a hammer, 
on the back of a wedge, is incomparably greater than any mere 
pressure, and will overcome it, HF oy 
Bias eo OF 


THE SCREW. 167 


OF THE SCREW. 


200. THE Screw is one of six mechanical powers, chiefly 
used in pressing or squeezing bodies close, though sometimes 
also in raising weights. 

The screw is a spiral thread or groove cut round a cylin- 
der, and every where making the same angle with the length 
of it. So that if the surface of the cylinder, with this spiral 
thread on it, were unfolded and stretched into a plane, the 
spiral thread would form a straight inclined plane, whose 
length would be to its height, as the circumference of the-cy- 
linder, is to the distance between two threads of the screw : 
as is evident by considering that, in making one round, the 
spiral rises along the cylinder the distance between the two 
threads. : . | 


PROPOSITION XXXVII. 


201. The Force of a Power applied to turn a Screw round, is 
to the Force with which it presses upward or downward, set- 
ting aside the Fraction, as the Distance between two Threads, 
as to the Circumference where the Power is applied. 


THE screw being an inclined plane, or half wedge, whose 
height is the distance between two threads, and its base the 
circumference of the screw; and the force in the horizontal 
direction, being to that in the vertical one, as the lines per- 
pendicular to them, namely, as the height of the plane, or — 
distance of the two threads, is to the base of the plane, or 
circumference of the screw ; therefore the power is to the 
pressure, as the distance of two threads is to that circumfe- 
- rence. But, by means of a handle or lever, the gain in power 
is increased in the proportion of the radius of the screw to 
the radius of the power, or length of the handle, or as their 
circumferences. Therefore, finally, the power is to the pres- 
sure, as the distance of the threads, is to the circumference 
described by the power. © 

202. Corol. When the screw is put in motion; then the 
power is to the weight which would keep it in equilibrio, as 
‘the velocity of the ihiter is to that of the former ; and hence 
their two momenta are equal, which are produced by multi- 
plying each weight or power by its own velocity. So that 
this is a general property in all the mechanical powers, namely, 
that the momentum of a power is equal to that of the weight 
which would balance it in equilibrio; or that each of them is 
reciprocally proportional to its velocity. | 


SCHOLIUM. 


168 MECHANICS. 
SCHOLIUM. 


203. Hence we can easily ses 
compute the force of any ma- i Gal 
chine turned by a screw. Let. = 
the annexed figure representa {iif 
press driven by ascrew, whose 
threads are each a quarter of 
an inch asunder; and let the 
screw be turned by a handle 
of 4 feet long, from a to 8 ; 
then, if the natural force of a aff 
man, by which he can lift, pull, HULURAGATAT RACER ETAL 
or draw, be 150 pounds ; and it be required to determine with 
what force the screw will press on the boad at p, when the 
man turns the handle at a and 8, with his whole force. Then 
the diameter az of the power being 4 feet, or 48 inches, its 
circumference is 48 X 3°1416 or 1504 nearly ; and the dis- 
tance of the threads being 1 of an inch ; therefore the power 
is to the pressure as 1 to se : but the power is equal to 
1501b ; theref. as 1; 6031 : : 150: 90480; and consequently 
the pressure at p is equal toa nr of 90480 pounds, inde- 
pendent of friction. 

204. Again, if the end- 
less screw az be turned by 
a handle ac of 20 inches, 
the threads of the screw 
being distant half an inch 
each ; and the screw turns 
a toothed wheel ©, whose 
pinion x turns another 
wheel Fr, and the pinion m 
of this another wheel c, to 
the pinion or barrel of 
which is hung a weight w ; 
it is required to determine 
what weight the man will 
be able to raise, working at 
the handle c ; supposing 
the diameters of the wheels 
to be 18 inches, and those 
of the pinions and barrel 2 
‘inches ; the teeth and pin- 
ions being all of a size. 





Za ; 
nt 
el 





Here 


CENTRE OF GRAVITY. 169 
* 

Here 20X3: 1416 X2=125-664, is the circumference of 
the power. 

And 125°664 to 4 or 251-328 to 1, is the force of the screw 
alone. ae 

Also, 18 i 2, or 9 to 1, being the proportion of the wheels 
to the pinions ; “and a6 there are three of them, therefore 9° 
to 15, or 729 to 1, is the power gained by the wheels. — 

Consequently 251- 328X729, to 1, or 183218} to 1 nearly, 
is the ratio of the power to the weight, arising “fom the ad: 
vantage both of the screw and the wheels. 

But the power is 150lb; therefore 150 X 1832181, or 
27 482716 pounds, is the. weight the man can sustain, which is 
equal to 12269 tons weight. 

But the power has to overcome, not only the weight, but 
also the friction of the screw, which is very great, in some 
cases equal to the weight itself, since it is sometimes sufficient 
to sustain the weight, when the power is taken off. 


ON THE CENTRE OF GRAVITY. 


205. THE Centre of Gravity of a body, is a certain 
point within it, on which the body being freely suspended, it 
will rest in any position ; and it will always descend to the 
lowest place to which it can get, in other positions. 


PROPOSITION XXXVII. 


206. If a Perpendicular to the Horizon, from the centre of 
Gravity of any body, fall within the Base of the Body, it will 
gest in that Position; but of the Perpendicular fall without 
the Base, the Baer, will mot rest in that Position, but will 
tumble down. ° 
For, if cz, be the perp. 

from the centre of gravity 

c, within the base : then the 

body cannot fall over towards - 

a ; because, in turning on the 

point a, the centre of gravity » 

c would describe an arc 

which would rise from-c to 

E5 contrary to the nature of that centre. Pick only rests 

when in the lowest place. For the same reason, the body 

. will not fall towards p. ‘And therefore it will stand in that 


position. y 
Vor. It. pon 2S." But 

















170 J STATICS. 
& 


But if the perpendicular fall without the base, as cb; then 
the body will tumble over on that side: because in turning 
on the point a, the renite ¢ descends by describing the de- 
scending arc ce. 

207. Corol. 1. If a aE eer drawn from the betes 
of gravity, fall just on the extremity of the base; the body 
may stand; but any the least force will cause, it to fall that 
way. “And the nearer the perpendicular is to any side, or 
the narrower the base is, the easier it will be made to fall, or 
be pushed over that way ; because the centre of gravity has 
the less height to rise: which is. the reason that a globe is 
made to roll on a smooth plane by any the least force. But 
the nearer the perpendicular is to. the middle of the base or 
the broader the base’is, the firmer the body stands. 

208. Corol. 2. Hence if the centre of gravity of a body 
be supported, the whole body is supported. And the place 
of the centre of gravity must be accounted the place of the 
body: for into that point the whole matter of the body may 
be supposed to be collected, and therefore all the force also 
with which it endeavours to descend. 

209. Corel. 3. From the property which 
the centre of gravity has, of always descend- — 
ing to the lowest point, is derived an easy 
~ echanical method of finding that centre. 


Thus, if the body be hung up by any 
point a, anda plumb line as be hung by the 
same point, it will pass through the centre 
of gravity ; because that centre is not in the 
lowest point till it fall in the plumb line. | 
Mark the line ap on it. hen hang the — 
body up by any other point p, with a plumb — 
line pe, which will also pass through the 
centre of gravity, for the same reason as’ 
before ; and therefore that centre must be _ 
at ¢ where the two plumb lines cross each 
other. 





210. Or, if the body. be. suspended by - 
two or more cords er, cu, &c. then a plumb is 
line from the point‘c will cut the body x mits 
centre of gravity c. 





211.. Like- 


CENTRE OF GRAVITY. 171 


O11. Likewise, ‘because a body .rests when its centre of 
gravity is supported, but not else ; we hence derive another 
ee method of finding that centre mechanically. For, if the 

ody be laid on the edge of a prism, or over one side of ta- 
he and moyed backward and forward till it rest, or balance 
itself; then is the centre of gravity just over the line of the 
edge. And if the body be then shifted into another position, 
and balanced on the edge again, this line will also pass by the 
centre of gravity ; and consequently the intersechan of the 
two will give the centre itself, 


PROPOSITION XXXIX. 


212. The common Centre of Gravity c of any two Bodies a, 8 
divides the Line joining their Centres, into two Parts, tohich 
are Recsprocally as the Bodies. 


Thatis, ac: Bes? BIA. 


For, if the centre of gravity c be supported, the two 
bodies a and B will be supported, ©- : 3 
and will rest in equilibrio. But ne (0°%. To 
by the nature of the lever, when 
two bodies ‘are in equilibrio about a fixed point ¢, they are 
reciprocally as their distances from that point; therefore 
A:B:4CB: CA 


213. Corol. 1. Hence ap: ac:: 4-+ 8:8; or, the whole 
distance between the two bodies, is to the distance of either 
of them from the common centre, as the sum of the bodies is 
to the other body. 


. 214, Corol. 2. Hence also, ca, a==cs . 83 or the'two pro- 
ducts are equal, which are made by multiplying each body by 
its distance from the centre of gravity. 


215, Corol. 3. As the centre cis pressed wilh a force equal 
to both the weights a and B, while the points a and: ‘Bare each 
pressed with the respective weights a ands. Therefore, if 
the two bodies be both united in their common centre c, wad 
only the ends a and 8 of the line as be supported, each will 
still bear, or be pressed by the same weights a and B as before. 
So that, if a weight of 1001b. be laid ona bar at c, supported 
by two men ata and g, distant from c, the one 4 feet, and the 
other 6 feet; then the nearer will bear the weight ‘of 60lb, 
and the farther only 40lb. weight. 

216. Corol. 


et 
aig 


i72 ye SAITO: 


216. Corol. 4. Since the = 
effect of any body to turn . Ea 7s ©)-@ 
a lever about the fixed , .§ ¥ HD por 
point c, is as that body and. A UB mi a 

as its distance from that point; therefore, if c be the common 
centre of gravity of all the bodies a, 3, p, x, F, placed in the 
straight line ar ; thenis,ca.a--ca.p=cp.p-+ce. E+ 
cr. F; or, the sum of the products on one side equal to the 
sum of the products on the other, made by multiplying each 
body by its distance from that centre. And if several bodies 
be in equilibrium on any straight lever, then the prop. is in the 
centre of gravity. | 

217, Corol. 5. And though 
the bodies be not situated in 
a straight line, but scattered 
about in any promiscuous man- 
ner, the same property as in the 
last corollary still holds true, 
if perpendiculars to any line’ 
whatever, af be drawn through - Wea ; 
the several bodies, and their common centre of gravity, name- 
ly, that ca: a-+ cb =cd.v+ce.e+cf.r.. For the 
bodies have the same effect on the line af, to turn it about the 

" point c, whether they are placed at the points a, b, d, e, f, or 
in any part of the perpendiculars aa, Bb, pd, re, Ff. 









PROPOSITION XL. 


218. Jf there be three. or ‘more Bodies, and tf a line be drawn 
from any one Body vp to the Centre of Gravity of the rest c ; 


then the Common Centre of Gravity & of all the Bodies, divides 


the line cp into two Parts inn, which are Reciprocally Pro- 
_ portronal as the Body p to the sum of all the other Bodies. 


That is, ce : ep::p:a+e&c.. 


For, suppose the bodies a ands, 
to be collected into the cemmon ¥% 
centre of gravity c, and let their sum 
becalleds. Then, by the last prop. 
CE2EDt: Dis orate... 





217. Corol. Hence we have a method of finding the com- 


‘mon centre of gravity of any number of bodies ; namely; by 
first finding the centre of any two of them, then the centre 
of that centre and a third, and so on for a fourth, or fifth, 
&c. ae 
PROPOSITION 


CENTRE OF GRAVITY. 173 


PROPOSITION XLL 


220. If. there be taken any Point p, in the Line passing through 
the Centres of Gravity of two Bodies ; then the sum of the two 
Products, of each Body multiplied by its Distance from that 
Potnt, is equai to the Product of the Sum of the Bodies multi- 
plied by the Distance of their Common Centre of ratty c 
from the same Point r. 


‘That i is, PA. A-- PB B = Prc.a-+e, 
For, by the 38th, ca. a = cp . B, 


that is, PA—PC. A= PC—PB. BS 3 
therefore by adding, . CP e P 


PA. A-+PB. B= Pc. a-+B. 











221. Corol, 1. Hence, the two bodies. and shave the 
same force to turn the lever about the point P, as if they were 
both placed ‘in c their common centre of gravity. 

_ Or, if the line, with the bodies, move about the point P; 
the sum of the momenta of 4 and s, is equal to the inereptamn 
of the sums, or a-+3 placed at. the centre c. 


{ 


222. Corol. 2. The samé is also true of any number of 
bodies whatever, as will appear by cor. 4, prop. 39, namely, 
pa. A+ PB.B+PpD.p &c. = Pc. at B+ D &c. where p 
ig In any point whatever in the line ac. 

And, by cor. 5, prop. 39, the same thing is true when the 
bodies are not placed in that line, but any where in the _per- 
pendiculars passing through the points 4, B, p, &c.; namely, 
Pa .a-+ eb. B+ Pd. p &e. =rpc.atsep-+p &c. 


223. Corol. 3. And if a plane pass through the Sainte ys per- 
pendicular to the line-cp; then the distance of the common 
centre of gravity from that plane, is 
PA.A-—-+ PB’. Bf Pd. p &c. 

“ade B+ pb &c.- 
of all the forces divided by the sum of all the bodies. Or, if 
A, B, D, &c. be the several particles of one mass or compound 
body ; ; then the distance of the centre of gravity of the body, 
below any given point p, is equal to the forces of all the par- 
ticles divided by the whole mass or body, that is, equal to all 


the pA. a, Pb. By Pd. pv, &c, siyided by the body or sum of 
particles a, 8, D, &c. 





r= , that is, equal to the sum 


PROPOSITION 


174 je STATICS. 


PROPOSITION XLIT, 


224. To fad the Centre of Grivily of any Body, or of any Sys- 
tem of Ree, 


“'Tarovan any faint P draw 








a plane, and let ra, pb, pd, &c. OB ; 

be the distance of the bodies P 2 ‘| 5 ke a Be 
A, B, D, &c. from the plane ; | hy iG ee 
then, by the last cor. the dis- = | | ue 
tance of the common centre of © A. @p F 





gravity from the plane, will be 
rome Atpb.s+pd.p &c. 
why A+ B+ 0 &c. 


295. Or, if é be any body, and qpr any plane ; draw paz 
ke: perpendicular to ex, and through a, B, &c. draw i innume- 
rable sections of the body 6 parallel ts pape 
to the plane er. Lets denote any v— R 
of these sections, and d = pa, or PB, 
&c. its distance from the plane gr. | 
Then will the distance of the centre 
of gravity of the body from the plane 

‘sum _of all the d’s | 
be Po = b ee, An d if ; 
the distance be thus found for two in- 
tersecting planes, they will give the 
point ; in which the centre is piace 














296. But the distance from one plane i is niiMelent, for. any 
regular body, because it is evident that in such a figure, the 
centre of gravity is in the axis, or line vee through the 
centres of all the parallel sections. 

Thus, if‘the figure be a parallelogram, or a 
cylinder, or any prism whatever ; then the axis. 
or line, or plane ps, which bisects all the sec- 
tions parallel to er, will pass through the 
centre of gravity, of all those sections, and 
consequently through that of the whole figure | 
c.. Then, all the sections s being equal, and 
the body b= rs . s, the distance of the centre 
will be pc = : 





PA. SPB. $+ ic. . PA+PB a PD aitiae ee PA -+ pp + Ke, 
b ate b a PS. 
But 











CENTRE OF GRAVITY. 175, 


But va + ps Ne &c. is the sum of an arithmetical pro: 
gression, beginning at 0, and increasing to the greatest term 
ps, the number of the terms being also equal to rs; there- 
fore the sum pa -+ pp -+ &c.=1 Ps . ps; and consequently 
ro ee = ips; that is, the centre of gravity is in the 

_ PSs | Ae 
middle of the axis of any figure whose parallel sections are 
equal. — . 3 


227. In other figures, whose parallel sections are not equal, 
but varying according to some general law, it will not be easy 
to find the sum of all the pa. s, pp. s', pp. s”, &c. except by 
the general method of Fluxions ; which case therefore will be 
best reserved, till we come to treat of that doctrine. _ It will 
be proper however to add here some examples of another me- 
thod of finding the centre of gravity of a triangle, or any other 
right-lined plane figure. 3 , 


PROPOSITION XLHI. 
_ 228.. To find the Centre of Gravity of a Triangle. 


From any two of the angles draw 
lines ap, cr, to bisect the opposite | 
sides, so will their intersection @ be — — 
the centre of gravity of the triangle. 

For, because’ ap bisects sc, it bi- 
sects also all its parallels, namely, all 
the parallel sections of the figure ; 
therefore ap passes through the cen- s 
tres of gravity of all the parallel sections or component parts 
of the figure ; and consequently the centre of gravity of the 
whole figure lies in the line ap. For the same reason, it also 
lies in the line ce. Consequently it is in their common point 
of intersection c. aise 





229. Corol. The distance of the point c, is ac = 2 ap, and 

cG =2cE; or aa= 2ep, and ce = 2cE. 
For, draw sr parallel to ap, and produce ce to meet it in F. 
Then the triangles arc, ser are similar, and also equal, because 
AE = BE; consequently ac = sr. But the triangles coc, car 
are also equiangular, and cz being = 2cp, therefore Br = 2cp. 
But Br is also = ac; consequently ac=2ep, or 2p. In like 
_ Manner, ce = cE or 2c. 
| | PROPOSITION 


176 | - STATICS. 
PROPOSITION XLIV. 


230. To find the Centre of Gravity us a a 


~ Divine the trapezium ABCD into . 

two triangles, by the diagonal gp, and 
find £, F, the centres of gravity of 
these two triangles ; then shall the 
centre of gravity of the trapezium lie 
in the line er connecting them. And 
. therefore if er be divided, in «, in the 
alternate ratio of the two triangles, 
namely, EG : Ge: : triangle scp : triangle agp, then ¢ will be 
the centre of gravity of the trapezium. 
. 231. Or, having found the two points £,.F, if the trapezium 
be divided into two other triangles pac, pac, by the other dia- 
gonal ac, and the centres of gravity # ‘and 1 of these two tri- 
angles be also found ; then the centre of gravity of the trapezium 
will also lie in the line ur. 

So that, lying in both the lines, er, ur, it must necessarily 
lie in their intersection ¢. 

232. And thus we are to proceed for a figure of any great- 
er number of sides, finding the centres of their component 
triangles and trapeziums, and then finding the common cen- 
. tre of every two of these, till they be all reduced into one 
only. 

Of the use of the place of the centre of gravity, and he 
nature of forces, the following practical problems are added ; 
viz. to find the force of a bank of the earth pressing against a 
wall and the force of the wall to support it ; also the push of 
e arch, with the thickness of the piers necessary to support 

t; also the ia ce and stress of beams and bars of timber 
a metal, &e. ie : 





PROPOSITION XLV. 


933.°: To deterinirie the Force with which a Bank of Earth, or 
such like, presses against a Wall, and the gasibancest of the 
Wall necessary to Support it. 


Let acpz be a vertical sectionofa G@-.A I BC 
‘bank of earth ; and suppose, that if it 
were not supported, a triangular part of 
it, aS ABE, would slide down, leaving it 
at what fs called the natural slope BE ; 
but that, by means of a wall aere, it is 
supported, and kept in its place.—It is 
required to find the force of ape, to 
slide down, and the dimensions of the 
wall arre, to support it. 





CENTRE OF GRAVITY. 177 


Let u be the centre of gravity of the triangle azz, through 
which draw. kui parallel to the slope face of the earth pr. 
Now the centre of gravity H may be accounted the place of 
the triangle ape, or the point into which it is all collected. 
Draw uu parallel, and xp perpendicular to az, alsé xx prep. 
to1xorse. Then if ux represent the force of the. triangle 
ABE inits natural direction HL, HK will denote its force in its 
direction Hk, and px the same force in the direction px per- 
pendicular to the lever ex, on whichit acts. Now the three 
triangles EAB, HKL, HKP are all similar ; therefore eB: Ba : : 


(HL : HK: :) w the weight of the triangle nas : = w, which 


will be the force of the triangle in the direction ux. Then, 
to find the effect of this force in the direction px, it will be, 
. BAO BA aR |. 

a8 HK : PK: : EB: AB: :—wW: -—w, the force at Kk, in 

EB EB? | | 

direction Px, perpendicularly on the lever ex, which is equal 
to tar. But tae. ap is the area of the triangle ase ; and if 
m be the specific gravity of the earth, then } ar . aB . mis as 
its weight. Therefore - 


EA . AB BA? :,-AB® © , ; cee 
Sear SOT a the force acting at x in di- 
: , 











rection px. And the effect of this pressure to overturn the 
the wall, is also as the length of the lever xz or }ar* : con- 


EA® . AB?» ouee 
—--,—m, for the perpendicular force 


sequently its effect i 
quen es eflect 18 Gun 





* The principle now employed in the solution of this 45th prop. is a little dif- 
ferent from that formerly used ; viz. by considering the triangle of earth age as 
acting by lines-1x, &c. parallel to the face of the slope Bx, instead of acting in 
directions parallel to the horizon az ; an alteration which gives the length of the~ 
lever Ex, only the half of what it was in the former way, viz. EK == % AE instead 
of 2 ax: but every thing else remaining the same as before. Indeed this prob- 
lem has formerly been treated on a variety of different hypotheses, by Mr. Mul- 
ler, &c. in this country, and by many French and other authors in other countries. 
And this has been chiefly owing to the uncertain way in which loose earth may 
be supposed to act in such a case; which on account of its various circumstances 
of tenacity, friction, &c. will not perhaps admit of a strict mechanical certainty. 
On these accounts it seems probable that it is to good experiments only, made on 
different kinds of earth and walls, that we may probably hope for a just and 
satisfactory solution of the problem. 

The above solution is given only in the most simple case of the problem. But 
the same _ ae may easily be extended to any other case that may be required, 
either in theory or practice, either with walls or banks of earth of different figures, 
and in different situations. 


You. I. 24 against, 


i786 STATICS. ' 


against , to overset the wall arc. Which must be balanced 
by the counter resistance of the wall, in order that it may at 
least be supported. 


Siw; if 'm be the centre of gravity of the wall, into whicli 
its whole matter may be supposed to be collected, and acting 
- inthe direction mw, its effect will be the same as if a weight 
w were suspended from the point n of the lever rn. Hence, 
if a be put for the area of the wall arrc, and n its specific 
gravity; then a. will be equal to the weight w, and a. 
n.. EN its effect on the lever to prevent it from turning about 
the point ». And as this effort must be equal to that of the 
triangle of shag that it may just support it, which was before 


. AB? AE? . AB? 
_——_m ; therefore a.m. FN= —M, 
a 6EB? 








found equal to = 
in case of an equilibrium. 


234. Butnow, both the breadth of the wall re, and the 
lever FN, or place of the centre of gravity m, will depend on 
the figure of the wall. If the wall be rectangular, or as 
broad at top as bottom; then rn =1 re, and the area a = 
AE . FE; consequently the effort of the wall a. .¥N is = 
eete £et -m, the effort of 

6EB? 
the earth. And the resolution of Par sapance gives the 


breadth of the wall re an 





$FE?; AE .2; which must be = 





—, drawi 
EB Si eV, ceitee 


perp. to rs? So that the breadth ‘of the wall is always pro- 
portional to the prep depth ae of the triangle ane. But the 
breadth must be made a little more than the above value of it, 
that it may be more than a bare balance to the earth.—If the 
angle of the slope & be 45°, as it is nearly in most cases ; 


AE Om m m 
eg fl oes atta ~-+ 5 —_— ‘ 
, then FE = Pa” in AE4/ gAE,/— very nearly, 
235. If the wall be of brick, its specific gravity is about 
2000, and that of the earth about 1984 ; namely, m to n as 1984 
to 2000 ; or they may be taken as equal ; then Vv = 1 very 


nearly ; and hence re = j;AE, or 2aE cael); That is, 
whenever a brick rectangular wall is made to support earth, 
its thickness must be at least 2 or 4; of its height. But if 

_ the 


CENTRE OF GRAVITY. 178 


the er be of psi whose apesiing gravity is about 2520 ; 
then — = 4, and v — = 4/4 = 895; hence re = -358 az 


oo. 5 aan that i is, whed the rectangular wall is of stone, the 
breadth must be at least 55, of its height. 
936. But if the figure of the wall 
be a triangle, the outer side tapering « 
toa pointattop. Then the lever ry. 
= 2re,and the areaa = 1 FE. AE; 
consequently its effort a.m . FN is = 
i FE? .aE.”; which being dike = 
AES, AB? 
6BE? 
ae V = =10/ 5 — ~ for the breadth 


of the bor sf the atop for an equilibrium in this case also. 





m, the equation gives Fr = 





—If the angle of the slope & be 45°; then will rr be = ae 
/> — =1aE J. And when this wall is of brick; then Fr 


=e nearly. But when it is of stene; then} f— era 


s= 4 nearly ; that i is, the triangular stone wall must es its 
thickness at bottom equal to 4 of its height. And in like man- 
ner, for other figures of the ‘wall and also for other he of 
the earth. | o* 


PROPOSITION XLVI. 


237. To determine the Thickness of a Pier, necessary to sup- 
port a given Arch. 


Let azpcp be half 
the arch, and pDEFG 
the pier.- From the 
centre of gravity x of 
the half arch draw ku 
perp. oa; also oxr, 
and TK@P perp. to it ; 
also draw ne and ai 
perp to TP, or paral- 
Jeltooxr. Then if xx 
represent the weight 
of the arch sepa, in the direction of gravity, this will resolve 
into Kq, the force acting against the pier perp. to the joint sr, 
and 1@ the part of the force parallel to the same. Now ke 

denotes 





180 ' STATICS. 


denotes the only force 
acting perp. on the arm 
Gp, of the crooked le- 
ver Fep, to tur the — 
pier about the point c ; 
conseg. Ka X ep will 
denote the efficacious. 
force of the arch to o- 
verturn the pier. 
Again, the weight of 
the pier is as the area | 
pF XFc ; therefore pF . rc . 1¥G, oripF . Fe?, is its effect on 
the lever irc, to prevent the pier from being overset ; sup- 
posing the length of the pier, from point to point, to ben 
more than the thickness of the arch. 
But that the pier and the arch may be in equilibrio, these 
two efforts must be equal. Therefore we have lor . r¢?= 
KQ.GP.A 


KL 
thickness of the pier ra ; a denoting the area of the half arch 
BCDA™. : Lee 

Example 1. Suppose the arc asm to be a semicircle ; and 
that cp or oa or op==45, sc = 7 feet, ar=20. Hence av = 
52, pF=cu=72. Also by measurement are found ox=50°3, 
KL=40°6, Lo=29°7, To=30°87, Kg=24, the area Bcpa==750 
=a: and putting re=z the breadth of the pier. 

Then te=To-+ ve = 30°87 + 2, and KL: LO: : TE: EV 
= 22°58-+-0°732, 

then ce —rv=cy=49°-42—°732, 

lastly ok : KL : : GV: GP==39°89— 59x. 

These values being now substituted in the theorem ipr. 


KQ — . cg give 2672 = 17665 at 261-bx, or x? -[- 











, an equation, by which will be determined the 


rq?2= 








* Note. As itis commonly a troublesome thing to calculate the place af the 
centre of gravity x of the half arc apcs, it may be easily, and sufficiently near, 
found mechanically in the manner described in art. 211, thus: Construct that 
space apcB accurately by a scale to the given dimensions, on a plate of any 
uniform fiat substance, or ‘even card paper; then cut it nicely out by the ex- 
treme lines, and balance it over any edge or the sides of a table in two positions, 
and the intersection of the two places will give the situation of the point x ; then 
the distances or lines may be measured by the scale, except those depending on - 
the breadth of the pier re, viz. the lines as mentioned im the examples. . 


| 9° 26x 


STRENGTH AND STRESS OF BEAMS, &c. 181 


7-262 = 490-7; the root of which quadratic equation gives 
x = 18-8 feet = pk or FG, the thickness of the pier sought. 

Example 2. Suppose the span to be 100 feet, the height 40 
feet, the thickness at the top 6 feet, and the height of the pier 
to the springer 20 feet, as before. 

Here the: fig. 
may be considered 
as a circular seg- 
ment, having the 
versed sine oB = 
40, and the right 

sine oA or o¢ = 
50; also sp = 6, 
cr= 20, and aF = 
66. Now, by the 
nature of the. cir- 
‘cle, whose centre is w, the radius WB = 
oB? --oc? 407-50? 

i oe ee — = 51}; hence ow = 511 — 40=111; 
and the area of the semi-segment ozc is found to be 1491 ; 
which is taken from the rectangle opec = op. oc = 46 x 50 = 
2300, there remains 809 = a, the area of the space spec. 
Hence, by the methed of balancing this space, and measur- 
ing the lines, there will be found, kc = 18, 1K = 34:6, 1x = 
42, xx = 24, ox = 8, 1@ = 19:4, tre = 35°6, and ta = 
35°6 + a, putting « = en, the breadth of the pier. Then 
IK : KX {* TH :.HvV = 24°7 e O:'7xz; hence cH — Hv = 
41:3 — 0-7 = ev, and 1x : 1k: : GV : GP-= 34-02 — 0-582. 
These values being now substituted in the theorem ter. 


cdl! gives 33a2 = 15431:47 — 2532, or x? + 








Fg?= 








IK 
8x. = 467-62, the root of which quadratic equation gives x = _ 
18 = EH or FG, the breadth of the pier, and which is probably — 
very near the truth. 


LEORes 
ON THE STRENGTH AND STRESS OF BEAMS OR 
BARS OF TIMBER AND METAL, &. -” 


238. Another use of the centre of gravity, whith may be 
here considered, is in determining the strength and the 
stress of beams and bars of timber and metal, &c. in differ- 
ent positions; that is, the force or resistance which a beam 
_or bar makes, to oppose any exertion or endeavour made to 
break it: and the force or exertion tending to break it ; 

both 


162 STATICS, 


both of which will be different according to the place and pe- 
sition of the centres ie gravity. 


PROPOSITION XLVI. 


239. The Absolute Strength of any Bar in the Direction ef its 
Length, is Directly Proportional to the Area ¢ its Transverse 
Sectron. 


Suprose the bar to be suspended by one end, and hanging 
freely in the manner of a pendulum; and suppose it to be 
strained in direction of its length, by any force, or weight 
acting at the lower part, in .the direction of that length, sufli- 
cient to break the bar, or to separate all its particles. Now, 
as the straining force acts in the direction of the length all the — 
particles in the transverse section of the body, where it breaks, 
are equally strained at the same time ; and they must all sepa- 
rate or break together, as the bar is supposed to be. of uniform 
texture. Thus then, the particles all adhering and resisting 
with equal force, the united strength of the whole, will be 
proportional to the number of eB er as the transverse sec- 
tion at the fracture. | 

240. Corol. 1. Hence the various ip cite of bars backe' no 
difference in their absolute strength : this depending only on 
the area of the section, and must be the same in all equal 
areas, whether round, or square, or oblong, or solid, or hol- 
low, &c. 

241. Corel. 2. Hence also, the absolute strengths of differ- 
ent bars, of the same materials, are to each other as their 
transverse sections, whatever their shape or form may be. 

242. Corol. 3. The bar is of equal strength in every part 
of it, when of any uniform thickness, or prismatic shape, 
and is equally liable to be drawn asunder at any part of its 
length, whatever that length may be, by a weight acting at 
the bottom, independent of the weight of the bar itself; but 
when considered with its own weight, it is the more disposed 
to break, and with the less additional appended weight, the 
longer the bar is on account of its own weight increasing 
with its length. And, for the same reason, it will be more 
and more liable to be broken at every point of its length, all 
the way ‘in ascending or counting from the bottom to the 
top, where it may always be expected to part asunder. And 
hence we see the reason why longer bars are, in this way 
more liable to break than shorter ones, or with less ap- 
pended weights. Hence also we perceive that, by gradually 
increasing these weights, till the bar separates and is il 

then 


STRENGTH AND STRESS OF BEAMS, &c. 183 


ihen the last or greatest weight, is the proper measure of the 
absolute strength of the bar. And the same is the case with 
a rope, or cord, &c.—So much then for the longitudinal 
strength and stress of bodies. Proceed we now to consider 
those of their transverse actions. 


PROPOSITION XLVIII. 


243. The Strength of a Beam or Bar, of Wood or Metal, &c. 
an @ Lateral or Transverse Direction, to resist a Force acting 
- Laterally, is Proportional to the Area or Section of the Beam 
an that Place, Drawn into the Distance of its Centre of Gra- 

vity from the Place where the Force acts, or where the Frac- 
ture will end. 


. Let az represent the beam 
er bar, supported at its two 
ends, and on which is laida 
weight w, to cause a trans- 
verse fracture abee. The force 
w acting downwards there, the 
fracture will commence or open 
across the fibres, in the oppo- 
site or lowest line ab ; from | 
thence, as the weight presses down the upper line ee, the 
fracture will open more and more below, and extend gradual- 
ly upwards, successively to the parallel lines of fibres cc, dd, 
&c. till it arrive at, and finally open in the last line of fibres 
ee, where it ends ; when the whole fracture is in the form of 
a wedge widest at the bottom, and ending in an edge or line ee 
at top. Now the area ae contains and denotes the sum of all 
the fibres to be broken or torn asunder ; and as they are sup- 
posed to be all equal to one another, in absolute strength, that 
area will denote the aggregrate or whole strength of all the 
fibres in the longitudinal direction, as in the foregoing propo- 
sition. But, with regard to lateral strength, each fibre must 
be considered as acting at the extremity of a lever whose cen- 
tre of motion is in the line ee: thus, each fibre in the line ab, 
will resist the fracture, by a force proportional to the prodact 
of its individual strength into its distance ae from the centre 
of motion, consequently the resistance of all the fibres in ad, 
will be expressed by ab Xae. In like manner, the aggregate. 
resistance of another course of fibres, parallel to ab, as cc, 
will be denoted by cc Xce ; and a third, as dd by dd Xde ; and 
_ so on throughout the whole fracture. So that the sum of all 
these products will express the total strength or resistance of 





al 


184 ey STATICS. 


all. the fibres or of the beam in that part. But, by art. 222, 

the sum of all these products.is equal to the product of the 

area aeeb, into the distance of its centre of gravely ‘from ee. 
. Hence the proposition is. manifest. 


244. Corol. 1, Hence it is ‘evident that the lateral strength 
of a bar, must be considerable less than the absolute longitu- 
dinal strength considered in the former proposition, and will 

be broken by a much less force, than was there necessary to 

draw the bar asunder. lengthways. Because, in the one case 
the fibres must be all separated at once, in. an instant ; but in 
the other, they are overcome and broken successively, one 
after another, and in some portion of time. For instance, 
take a walking stick, and stretching it lengthways, it will bear 
a very great force before it can he drawn asunder ; but again 
taking such a stick, apply the middle of it to the bended knee, 
and with the two hands drawing the end towards you, the stick 
is broken across by a small force. 


245. Corol. 2. In square beams, the lateral strengths are, 
as the cubes of the breadths or depths. 


246. Corol. 3. And in general, the lateral strengths of any 
bars, whose sections are similar figures, are as the cubes of 
the similar sides of the sections. 


‘ 


947. Corol. 4. In cylindrical bee the lateral strength 
are as the cubes of the diameters. | 


248. Corol. 5. In rectangular beams, the lateral strengths 
are to each other, as the breadths and square of the depths. 


249. Corol. 6. Therefore a joist laid on its narrow edge, is 
stronger than when laid on its flat side horizontal, in propor- 
tion as the breadth exceeds the thickness. Thus jif'a joist be 
_ 10 inches broad, by 21 thick, then it will bear 4 times more’ 
when laid on edge, than when laid flat. Which shows. the- 
propriety of the modern method of Hepa with very thin, 
but deep joists. 


250. Corol. 7. Ifa beam be fixed firoaly by one end into a 
wall, in a horizontal position, and the fracture be caused by a 
weight suspended at the other end, the process would be the 
same, only that the fracture would commence above, and ter- 
minate at the lower side ; and the prop. and all the corolla-, 
ries would still hold good. 
251. Corol. 8. When acylinder or prism is made hollow, it. 

is stronger than. when solid, with an equal quantity of mate- 
riaks 


STRENGTH AND STRESS OF BEAMS, &. 185 


rials and length, in the same proportion as its outer diameter 
_ isgreater. Which shows the wisdom of Providence in mak- 
ing the stalks of corn, and the feathers and bones of animals, 
&c. tobe hollow. Also, if the hollow beam have the hollow 
or pipe not in the middle, but nearest to that side where the 
fracture is to end, it will be so much the stronger. 

252. Corol. 9. If the beam be a triangular prism, it will 
be strongest when laid with the edge upwards, if the fracture 
commence or open first on the under side; otherwise with 
the flat side upwards ; because in either case the centre of 
gravity is the farther from the ending of the fracture. And 
the same thing is true, and for the same reason, for any other 
shape of the prism. On the same account also, a square beam 
is stronger when laid, or when acting angle-wise, than when 
ona flat side. | 


PROPOSITION XLIX, 


253. The Lateral Strengths of Prismatic Beams, of the same 
materials, are Directly as the Areas of the Sections and the 
Distances of their Centres of Gravity; and Inversely as their 
Lengths and Weighis. 


Let ap and cp represent the 
two beams fixed horizontally, by 
their ends, into an upright wall 
ac. Now, by the last prop. the 
strength of either beam, con- 
sidered as without or indepen- 
dent of weight is as its section drawn into the distance of its 
centre of gravity from the fixed point, viz. as sc, where s de- 
notes the transverse section at a orc, and c the distance of 
its centre of gravity above the lowest point a orc. But the 
effort of their weight, w or w, tending to separate the fibres 
and break the beam, are, by the principle of the lever, as the 
weight drawn into the distance of the place where it may be 
supposed to be collected and applied, which is in the middle 
of the length of the beam; that is, the effort of the weight 
upon the beam is as wX1as. Hence the prop. is manifest. 

254. Corol. 1. Any extraneous weight or force also, any- 
where applied to the beam, will have a similar effect to break 
the beam as its own weight; that is, its effect will be as wX 

- d, as the weight drawn into the length of lever or distance 
_from’a where it is applied. 














Vor. I. 25 955. Corel, 


he FO STATICS. 


255. Corol. 2. When the beam is fixed at both ends, the 
same property will hold good, with this difference only, that 
in this case the beam is of the same strength, as another of an 
equal section, and only half the length, when fixed only at one 
end. For, if the longer beam were bisected, or cut in halves, 
each half would be in the same circumstances with respect to 
its fixed end, as the shorter beam of wane) length. 


256. Corol. 3. Square prisms and cylinders have their lat: | 
eral strengths proportional to the cubes of the depths, or di- 
ameters, directly, and to their lengths and weights inversely. 


Corol. 4. Similar prisms and cylinders have their strengths 
inversely proportional to their like linear dimensions, the 
smaller being comparatively larger in that proportion. For 
their strength increases as the cube of the diameter or of 
their length ; but their stress, from their weight and length of 
lever, as the 4th power of the length. 


257. Scholium. From the foregoing deductions it follows 
that, in similar bodies of the same texture, the force which 
tends to break them, or to make them liable to injury by ac- 
cidents, in the larger bodies, increases in a higher proportion 
than the force which tends to preserve them entire, or tg se- 
cure them against such accidents ; their disadvantage, or ten- 
dency to break by their own weight, increasing in the.same 
proportion as their length increases ;: so that, though a smaller 
beam may be firm and secure, yet a large and similar one may 
be so long as to break by its own weight. Hence, it is justly 
concluded, that what may appear very firm and successful in 
a model or small machine, may be weak and infirm, or even 
fallin pieces by its own weight, when it is executed on large 
dimensions according to the model. 

For, in similar bodies, or engines, or in animals, the greater 
must be always more liable to accidents, than the stwaller, and 
have a less relative strength, that is, the greater have not a 
strength in so great a proportion as their magnitude. A great- 
er column, for instance, is in much more danger of breaking 
by a fall, than a similar smaller one. A man is in more dan- 
ger from accidents of this kind than a child. An insect can 
bear and carry a load many times heavier than itself; where- - 
as a larger animal, as a horse, for instance, can hardly support 
a burden equal to ‘his own weight. 

From the same principle it is also justly infeed, that 
there are necessarily limits in all the works of nature ‘and 

art, 


STRENGTH AND STRESS OF BEAMS, &. 187 


art, which they cannot surpass in magnitude. Thus, for in- 
stance, were trees to be of a very enormous size, their 

branches would break and fall off by their own weight. Large 
animals have not strength in proportion to their size: and if 
there were any land animals much larger than those we know, 

they would hardly be able to move, and wate be perpetually 
subjected to most dangerous accidents. 

‘As to the sea animals indeed, the case is different, as the 
pressure of the water ina great measure sustains them ; ; and 
ey we find they are vastly larger than land animals. . 

From what has been said it clearly follows that to make 
fodies or engines, or animals, of equal relative strength, the 
larger ones must have grosser proportions, or a higher de- 
gree of thickness, than they have of length. And this senti- 
ment being suggested to us by continual experience, we natu- 
rally join the idea of greater strength and force with the gross- 
er proportions, and of agility with the more delicate ones. In 
architecture, where the appearance of solidity is no less re- 
garded than real firmness and strength, in order to satisfy a 
judicious eye and taste, the various orders of the columns 
serve to suggest different ideas of strength. But by the 
same principle, if we should suppose animals vastly large, 
from the gross proportions a heaviness and unwieldiness would 
arise, which would. make them useless to themselves, and dis- 
agreeable tothe eye. In this,as in all other cases, whatever 
generally pleases taste, not vitiated by prejudice of educa.ion, 
or by fabulous and marvellous relations, may be traced till it 
abprate to have a just foundation in nature. 


PROPOSITIONL. > 


258. If a Weight be placed, or a Force act, on any part of a 
Horizontal beam, supported at both ends, the Stress upon that 
part will be as the Rectangle or Produet of its tive Distances 
_ from the ee sake ends. 


Past is, the stress upon the 
beam as, at c, by the weight w, 
isasac X Bc. For, by the na- 
ture of the lever, the effect of 
the weight w, on the lever ac, 
is AC. W; a the effect of this 
force acting at c, on the lever 
pc, 1S AC. W. BC == ac.BC. W, 
And, the weight w being given, the effect or stress is as AC . 
Hed © 





259. Corol. 


188 ie STATICS. 


259. Corol. 1. 'The greatest stress is when the weight w is 
at the middle: for then the rectangle of the two halves, 
AC . AC = Jap tam = 1}ap?, is the greatest. And from the 
middle point, the stress is less and less all the way to the ex- 
tremities a and B, where it is nothing. 


260. Corol. 2. The same thing will obtain from the igen 
of the beam itself, or from any other weight diffused equally 
all over it; the stress in this case being the half of the 
former, So that, in all structures, we should avoid as much 
as possible, placing weights or strains in the middle of beams. 


261. Corol. 3. If w be the greatest weight that a beam can 
sustain at its middle point; and it be required to find the 
place where it will support any. greater weight w ; that point 
will be found by making, as w:w::1aB. Las, or 1aB2 : 
AC . BC Or Ac X (AB—AC) = AB. Ac — act. 


PROPOSITION LI. 


262. When a Beam is placed aslope, ats Strength in that oii. 
as to its Strength when Horizontal, to resist a Vertical Force, 


as the Square of Radius ts to nue Square of the Cosine of the 
Elevation. 


Let as be the beam standing aslope, 
cF perp. to the horizon are; then ep 
is the vertical section of the beam, and 
CE, perp. to as, is the transverse sec- 
tion, and is the same as when in the 
horizontal position. Now, the strength, 
in both positions, is as the section drawn 
into the distance of its centre of gravity < . 
from the point c. But the sections, be- A EF G 
ing of the same breadth, are as their 
depths, ep, ce; and the distances of the centres of gravity 
are as the same depths ; therefore the strengths are as cp . cp 
to cE . cE, or cp? to ce?. But, by the similar triangles cp, 
AFD, itis cD : CE: : AD; AF, a8 radius to the cosine of the 
elevation. Therefore the oblique strength is to the trans- 
verse strength, as ap? to ar?, the square of radius to the 
square of the cosine of elevation. 





263. Corol. 1. The strength of a beam increases from the 
horizontal position, where it is least, all the way as it revolves 
to the vertical position, where it is: the greatest. oe 


STRENGTH AND STRESS OF BEAMS, &. 189 


PROPOSITION LIL 


264. When Beams. eb Aslope, or Obliquely, and sustaining 
Weights, either at the Middle Points, or in any other Similar 
Situations, or Equally Diffused over their Lengths ; the Strains 
upon them are Directly as the Weights, and the Lengths, and 
the Cosines of Elevation. 


For, by the inclined plane, the weight is to the pressure on 
the plane, as ac to ar, as radius to the cosine of elevation : 
therefore the pressure is as the weight drawn into the cosine 
of the elevation. Hence the stress will be as the length of 
the beam and this force ; that is, as the weight x nee x 
cosine of elevation. : 


265. Corol. 1. When thd lengths and weights of beams are 
the same, the: stress is as the cosine of elevation’: ; and it is 
therefore the greatest whenit lies horizontal. 


266. Corol. 2. In all similar positions, and the weights va- 
rying as the lengths, or, the beams uniform ; then the stress 
varies as the squares of the lengths. . 


267. Corol. 3. When the weights 
are equal, on the oblique beam aps, 
and the horizontal one ac, and Bc 
is vertical: the stress on both beams 
is equal. For, the length into the 
cosine of elevation is the same in 
both; or aB X cos. A= ac X ra- 
dius. 





268. Corol. 4. But if the weights on the beams vary as 
their lengths ; then the stress will also vary in the same ratio. 


269. Corel. 5. And universally, the stress upon any point 
of an oblique beam, is as the rectangle of the segments of the 
beam, and the weight, and cosine of inclination, Breet ys : and 
the eh inversely. 


PROPOSITION 


190 | : STATICS. 


PROPOSITION LI. 


270. When.a Beam ts to sustain any Weight, or Pressure, or 
Force, acting Laterally; then the Strength ought to be as the 
Siress upon it; that is, the Breadth multiplied by the Square 
of the Depth, or in stmilar sections, the Cube of the Diaineter, 
an every place, ought to be proportional to the Length drawn 
into the Weight or Force acting on it. ‘And the same is true 
of several Different Preces of timber compared together. 


For every several piece of timber or metal, as well as 
every part of the same, ought to have its strength propor- 
tioned to the weight, force, or pressure it is to support. vAnd 
therefore the strength ought to be universally, or in every 
part as the stress upon it. But the strength is as the breadth 
into the square of the depth ; and the stress is as the weight 

or force into the distance it acts at. Therefore these must be 
in constant ratio. ‘This general property will give rise to the 
effect of different shapes in beams, according to particular 
circumstances ; as in the following corollaries. 


971. Corol. 1. If apc be a hori-~ 
zontal beam, fixed at the end ac, 
and sustaining a weight at the other 
end s. And if the sections at all 
places be similar figures ; and pE be 
the diameter at any place p; then 
BD will be every where as pE*. So that if avg be a right line, 
then Bec will be acubic parabola. In which case 2 of such 
a beam may be cut away, without any diminution of the 
strength.—But if the beam be bounded by two parallel planes, 
perpendicular to the horizon; then ep will be as ne? ; and 
then Bec will be the common parabolain which case a 3d part 
of the beam may be thus cut away. 

272. Corol. 2. Butif a weight press uniformly on every 
part of as ; and the sections in all points, as p, be similar ; 
then Bp? will be every where as pe? : and then Bec is the 
eeeobical parabola. Hh Et ; 





“But, in this disposition of the 
weight, if the beam be bounded by 
parallel planes,perpendicular tothe 
horizon ; then sp will be always as 
pE; and Bec a right line, or aBc a 
wedge. So that then half the beam 
may be cut away, without diminution of strength. 





273, Corol. 


STRENGTH AND STRESS OF BEAMS, &c. 191 


273. Corol. 3. If the beam az 
be supported at both ends; and 
if it sustain a weight at any vari-- 
able point p, or uniformly on all 
parts of its length; and if all the sections be similar figures ; 
then will the diameter pE? be every where as the rectangle 
AD . DB, — 

But if it be bounded by two parallel planes, perpendicular 
to the horizon ; then will pz? be every where as the rect- 
angle AD . Dr, and the curve AEB an ellipsis. 





274. Corol. 4. Butif a weight . 
be placed at any given point r, and 
all the sections be similar figures ; 
then will ap be as pr’, and ac, BG 
be two cubic parabolas. | 

But if the beam be bounded by two araiak planes, per- 
pendicular to the horizon; then ap is as pE?, and ac and Be 
are two common ee 





275. Scholium. The relative strengths of several sorts of 
wood, and of other badies;: as determined by Mr. Emerson, 
are as follow : 


Iron - = - - - > ~ 107 
Brass Beh ig yp ae we PL r - “ 50 
Bone ® - ; Q 
‘Box, View, Plumbiree, Oak ~ e ‘ 11 
Elm, Ash - - - =: - d $1 
Walnut, Thorn’- — - 72 


Red air, Holly, Elder, Plane; i rs ee Appletree. 7 
Beech, ebepsyared, Hazle - - 62 
Lead - a 3 he ae Ge 
Alder, Asp, Birch, White ay Willow at ee 6 
Fine freestone - — - - > - - 1 


A cylindric rod. of good clean fir, of 1 inch circumference, 
drawn lengthways, will bear at extremity 400 lbs; and a 
spear of fir, 2 inches diameter, will bear about 7 tons in that 
direction. 

‘A rod of good iron, of an inch circumference, will bear 
a stretch of near 3 tons weight. : 

A good hempen rope, of an inch circumference, will bear 
1000 lbs at the most. 

Hence Mr. Emerson concludes, that if a rod of fir, or of 

iron, 


192 STATICS. 


iron, or a rope of d inches diameter, were to lift 1 of the ex- 
treme weight; then | 

The fir would bear 84 d? hundred weights. 

The rope - - 22d? ditto. 

The iron - - 68 d? tons. 

Mr. Banks, an ingenious lecturer on mechanics, made many 
experiments on the strength of wood and metal ; whence he 
concludes, that cast iron is from 31 to 41 times stronger than 
oak of equal dimensions ; and from 5 to 61 times stronger 
than deal. And that bars of cast iron, an inch square, weigh- 
ing 9 lbs. to the yard in length, supported at the extremities, 
bear on an average, a load of 970 lbs. laterally. And they 
bend about an inch before they break. 

Many other experiments on the strength of different ma- 
terials, and curious results deduced from them, may be seen 
in Dr. Gregory’s and Mr. Emerson’s Treatises on Mechanics, 
as well ‘as some more propositions on the strength and stress 
of different bars. | 


oe 


ON THE CENTRES OF PERCUSSION, OSCILLATION, 
AND GYRATION. 


276. THE Centre of Percussion of a body, or asystem 
of bodies, revolving about a point, or axis, is that point, which 
striking an immoveable object, the whole mass shall not in- 
cline to either side, but rest as it were in equilibrio, without 
acting on the centre of suspension. 

277. The Centre. of Oscillation is that point, in a body 
vibrating by its gravity, in which if any body be placed, or if 
the whole mass be collected, it will perform its vibrations in 
the same time, and with the same angular velocity, as the 
whole body, about the same point or axis of suspension. 

278. The Centre of Gyration, is that point, in which if 
the whole mass be collected. the same angular velocity will 
be generated in the same time, by a given force acting at any 
place, as in the body or system itself. 

279. The angular motion of a body, or system of bodies, 
is the motion of a line conneeting any point and the centre or 
axis of motion; and is the same in all parts of the same re- 
volving body. And in different unconnected bodies each re- 
volving about a centre, the angular velocity is as the absolute 
velocity directly, and as the distance from the centre inverse- 
ly ; so that, if their absolute velocities be as their radii or 


distances, the angular velocities will be equal. © 
PROP- 


CENTRE OF PERCUSSION. 193 
PROPOSITION LIV. 


280. To ping the Centre of Percussion of a Body, or System 
of Bodies. 


Ler the bis revolve about an axis 

passing through any point s in the line 
sco, passing through the centres of gra- 
vity and percussion, Gc ando. Let my 
be the section of the body, or the plane » 
in which the axis sco moves. And 
conceive all the particles of the body to 
‘be reduced to this plane, hy perpendi- 
culars let fall from them to the plane: a 
supposition which will not affect the 
centres g, 0, nor the angular motion of 
the body. 





Let a be the place of one of the particles, so reduced ; 
join sa, and ray AP perpendicular to as, and aa perpendi- 
cular to sco; then apr will be the direction of a’s motion as 
it revolves about s; and the whole mass being stopped at o, 
the body a will urge the point p, forward, with a force pro- 
portional to its quantity of matter and velocity, or to its 
matter and distance from the point of suspension s ; that is, 
as a.sa; and the efficacy of this force in a direction. per- 
pendicular to so, at the point Pp, is as a .sa, by similar tri- 
angles; also, the effect of this force on the lever, to turn it 
ies o, being as the length of the lever, is as a. sa. Po= 
A.S *(so—sr) =a. sa.SO—A.SA.SP=A.Sa.S50 — 
a.SA2, In like manner, the forces of B and c, to turn the 
system about o, are as. | 

B.sb.S0—B. SB2, and 
c.sc .so—c:, so?, &c. 


But, since the forces on the contrary sides of o destroy one 
another, by the definition of this force, the sum of the posi- 
live parts of these quantities must be equal. to the sum oF the 
negative ee 
that is, a so +8.sb.so+c.sc.so > &e. = 

! ROS at etd ii sco? &c. ; and 
sa? +B. sp? +c. sc? &c. 
sats. sb+e . se &c.’ 
Vor. Il. 2600 which 





‘kos 
hence so. —=— 
A 


{94 ) > SPATICS. 


which is the distance of the centre of percussion: below the 
’ axis of motion. 


And here it may be observed that, if any of the points a, b, 
&c. fall on the contrary side of s, the corresponding product _ 
&. Sa, orB. sb, &c. must be made negative. 


281. Corol. 1. Since, by cor. 3, pr. 40, a-+s8-+c &c. 
or the body 6 X the distance of the centre of gravity, se, 
is=a.sa-+B.sb-+c.sc &c. which is the denominator 
of the value of so; therefore the distance of the centre of 
A.SA2 +8. sB2 +c. sc? &c. 


sc X body 6 


282. Corol. 2. Since, by Geometry, theor. 36, 37, 

it is sa? == se? ar GA? — 2sq . Ga, 

and sp? = sq? + cp? + Yse . ch, 

and sc? = sg? + gc? + Q@se . ac, &c. 3 
and, by cor. 5, pr. 40, the sum of the last terms is sothinie, 
namely, — sc . ca + 2sc . ch + Ysa . cc &c. = O; 
therefore the sum of the others, or a. sa? +.B. sB? &c. - 
is =(a + age se? +a.ca?-+B.cB2? +c. ec? &e. 
or =" 6.sc2 +a. ca? +.B. cB? +e. Ge? &c; 
which being substituted i in the numerator of the foregoing va- 
lue of So, oe | 

b.sc? +a. Ayhayd GB? + &c. 


, ‘e 
percussion, is so = 


so = 











9 


6 .sG 
A.GaA2 B.GB2 + c¢. Gc? &c. 
or SO = 5G + Se 
. SG 


~ 


283. Corol. 3..Hence the distance of the centre of pereus- 
sion always exceeds the distance of the centre of gravity, and 
ca? +B. GB? &c. 

b. se : 

| > a. Ga? + B. GB? &c. 

84. -. Go =—— ————— } 
2 And hence also, sc . Go as eS sat ; 


that 1 18 SG, GO Is always the same constant quantity, where- 
ever the point of suspension s is placed ; since the point ¢ 
and the bodies a, 8, &c. are constant. Or co is always 
‘reciprocally as sc, that is Go is less, as s¢ is greater ; and 
consequently the point oO rises upwards and approaches to- 
wards the point c, as the point s is removed to the greater 
distance ;. and they coincide when sc is infinite. But when s 
coinentes with c, then co.is infinite, or ois at an infinite dis- 
tance. ae : 


Ai. 
the excess is always co = 


PROPOSITION 


CENTRE OF PERCUSSION. 198 


PROPOSITION LY: 
‘O85. if a Body ‘A, at the Distance sa from an axis E paar 
through. s, be made to revolve about that axis by any Force 
acting at F in the Line sp, Perpendicular to the Aazs of Mo- 
tion: It 1s required to determine the Quantity or Matter of 
another Body Q, which being placed at p, the Point where the 
Force acts, it shall be accelerated in the Same Munner, as 
when a revolved at the Distance sa; and consequently, that 
the Angular Velocity of a and e about s s, may be the Same in 

* Both Cases. 
eee the nature of the lever, sa: sp: : f : 


—. ~.f, the effect of the force f, acting at P, A. 
on i ie body at a; that is, the force facting at 7 
Pp, will ae the same effect on the body a, as 

she 


the force ~ e nif acting directly at the point a. 


But as asp revolves altogether about the axis at s, the abso- 
lute velocities of the points a and s, or of the bodies a and 
@, will be as the radii sa, sp, of the circle described by them. 
Here then we have two Se a and @, which being urged 


directly by the forces f and ~ = a acquire velocities which are 


as sp and sa. And since the motive forces of bodies are as 





their mass and velocity: therefore ~ - > =: dae 

on Rat sa3 

re cc cure a odor hd Dn Carat Ga aaa > As 
SP 


which therefore expresses the mass of matter which, being 
placed at p, would receive the same angular motion from the 
action of any force at p, as the body a receives, So that the 
resistance of any body a, to a force acting at any point p, is 
directly as the square of its distance sa from the axis of mo- 
tion, and reciprocally as the ct ar te of the distance sp of the 
point where the force acts. 

286. Corol. 1. Hence the force whic accelerates the point 


SP 
p, is to the force of gravity, as te to 1, or ae sp? to 


A. SA. . 

. 287. Corol. 2. If any number of bodies 
A, B, ¢, be put in motion, about a fixed — 
axis passing through s, by a force act-— 
ing at P; the point p will be accelerated 
in the same manner, and consequently the 
_ whole system will have the same angular 

velocity, if instead of the bodies a, B, c, 





196 STATICS. 


placedat the distances sa, sz, sc, there be substituted the bodies 
wat cpt Se8 age ba ye 

| ae gar eres these being collected into the point r. 
And hence, the moving force being f, and the matter moved 
-. ® a. ga2+B. SBX+c. sc? fi. sp? 

eing ——__—_____ —__——+~—_~—_; theref. ———__->—________ 
tit 5 sp? : nek a.sa?-+B.SB2-++-¢.8c? 
is the accelerating force ; which therefore is to the accelerat- 
ing force of gravity, asf. sp? to a ..sa?-+B .sB?+4-c . sc?. 


288. Corol. 3. The angular velocity of the whole system 
Js SP 


velocity of the point p, is as the accelerating force, or di- 
rectly as the motive force f, and inversely as the mass 
A. SA? Xe. 








: but. the angular velocity is as the absolute velo- 
SP , 


city directly, and the radius sp inversely ; therefore the an- 

gular velocity of p, or of the whole sy.tem, which is the same 
eA, . SB | 

thing is as sidisct cate LIM wad bth d 

; A. SA?--B.. sB?-+c. sc? 


PROPOSITION LVI. 


289. To deterinine the Centre of Oscillation of any Compound 
Mass or Body mn, or of any System of Bodies a, B, c, Sc. 


Let mn be the plane of vibration, to which let all the 
matter be reduced, by letting fall perpendiculars from every 
particle, to this plane. Let 
@ be the centre of gravity, - 
and o the centre of oscilla- 
tion; through the axis s 
draw sco, and the horizon- 
tal line sq7; then from every 
particle, a, 8, c, &c. let fall 
perpendiculars aa, ap, Bb, Bq, 
cc, cr, to these two lines ; and 
jein sa, sB, sc; also, draw | 
Gm, on, perpendicular to sq. 
Now the forces of the weights 
A, B, Cc, to turn the body , 
about the axis, are a. Sp, B. 
sg,—e . sr; therefore, by 
cor. 3, prop. 55, the angular 





motion 


CENTRE OF OSCILLATION. 197 


A.sp-+B.sq—c sp 
A.SA2--B.SB2-+¢ . s¢8 
Also ; the angular veloc. any particle p, placed in,o, gene- 


«i S7 Sn ~. sm 
P Or 4.5 








motion generated by all these forces is 








‘in the system, by its weight, is ~ 
rates in y , by its weight, 1 52 502 a is 
because of the similar triangles sem, son. But, by the prob- 
lem, the vibrations are performed alike in both cases. and 
therefore, these two expressions must be equal to each other, 

et 8 .8g—c. sr 
that is, ie 2 ER EY 

SG.SO ~A.SA2-+-B. sp2-+ ec. sc? 
sm. A. SA? +B.SB2? +c. sc? 
oo eine dé: ie 
SG gore et ape mimie are 
But, by cor. 2, pr. 41, the sum a .sp+B.sq—c .sr=(a+p 
+c) . sm; fhevetire the penance Sos « ete ead a 


A. SA7 TB. sp? +6. sc? Ceeaa ai nana ine ware 


Ha ee ee Sere ae Fe ee a ae Nee vase ee SEI UES OSE Eat SRS a Ln Le 


Nala (a+ B B+ ¢) see ayer meee sec 
by prop. 42, which is the distance of the centre of oscillation 
0, below the axis of suspension; Where any of the products 
A. Sa, B.8b, must be. negative, when a, 6, &c. lie on the other 
side of s. 50 that this is the same expression as that for the 
distance of the centre of percussion, found in prop. 54. 

Hence it appears, that the centres of percussion and of os- 
cillation, are in the very same point. And therefore the pro- 
perties in all the corollaries there found for the former, are 
to be here understood of the latter. 

290. Corol..1. If p be any particle of a body, b, and d its 
distance from the axis of motions; also c, o the centres of 
gravity and oscillation. Then the distance of the centre of 








ee and hence 














oscillation of the bedy, from the axis of motion, is - - ~ 


sum of all the pd? 
‘sa X the body d © 3 

291. Corol. 2. If b denote the matter in any compound 
body, whose centres of gravity and oscillation are c and o; 
the body p; which being placed at pe, where the force acts as 
in the last proposition, and which receives the same motion 


from that force as the compound body 8, is rp = —— ob. 
Nig ie by corol. 2, prop. 54, this body Pis = - = - - 
sat + Bes ss? +c. sc? sc? 
gp? 
s¢.SsO.b =a. sa? + B. sp? + c.sc® . therefore ep = 
G . S80 
s .b. 
SP 


30 == 














But, by corol. 1, prop. 53, 





SCHOLIUM. 


198 STATICS. 


SCHOLIUM. 


- 999. By the mettiad of. ee ; the centre of oscilbation 
for a regular body, will be found from cor. 1. But for an 
irregular one ; suspend it at the given point; and hang up 
also a simple pendulum of such a length, that making them 
both vibrate, they may keep time together. Then the length 
of the simple pendulum, is equal to the distance of the cen- 
tre of oscillation of the body, below the point of suspension. 


293. Or it will be still better found thus: Suspend the 


body very freely by the given point, and make it vibrate in - 


small arcs, counting the number of vibrations it makes in any 
- time, asa minute, by a good stop watch ; and let that number 
of vibrations made in a minute be called x: ‘Then shall the 
140850 

inches 

nn 
For, the length of the pendylum vibrating seconds, or 60 times 
in a minute, being 391 inches ; and the lengths of pendulums 
bemg reciprocally as the square of the number of vibrations 
602 X 392 
nn 





distance of the centre of oscillation, be so = 


made in the same time ; therefore n? : 607 : : 391: 


a os : the length of the pendulum which vibrates rs 


_ times ina ‘minute, or the distance of the centre of oscillation 
below the axis of motion. 


294. The foregoing determination of the point, into which | 


all the matter of a body being collected, it shall. oscillate: in 
the same manner as before, only respects the case in which 
the body is put in motion by the gravity of its own particles, 
and the point is the centre of oscillation : but when the body 
is put in motion by some other extraneous force, instead of 
its gravity, then the point is different from the former, and is 
called the Centre of Gyration ; which ig determined in the 
following manner : : 


‘ a 
PROPOSITION 


CENTRE OF GYRATION. 199 
* PROPOSITION LVIL 


295. To Heteniiine the Centre of Gyration of a Compound Body 
Cyr yy er. of. System of Hades 


Ler n be the centre of gyration, or. 
the point inte which all the particles a, 
B, c, &c. being collected, it shall receive 

the same angular motion from a force f 
acting at r, as the whole system re- 
ceives. 

-. Now, by cor. 3, pr. 54, the angular 
velocity generated in the system by the 


. SP aa 
nn ee nN 
, sa2 + B. SB? &c.”. 


by the same, “the pa velocity of the system placed in R, is 


fi. sp 
Ca Ge aay : then, by making these two expres 
sions Hee to each vie the equation gives - - - 
ap .Sa2 +B. an? tc. sc? 
sk = = 4/- aa cue a 
centre of gyration below the axis of motion. 

296. Corol. 1. Because a . sa? + 8. sp? & c. =sc.so. 8, 
where c¢ is the centre of gravity, o the centre of weeillation. 
and 6 the body a-+8-+c &c. ; therefore sn?=sc . so; that 
is, the distance of the centre of gyration, is a mean propor- 
tional between those of gravity and oscillation. 

297. Corol. 2. If p denote any particle of a body 4, at d dis- 
sum ofall the e pd 


aghody bc. 


force f, is as- 





, for the distance of the 








‘tance from the axis of motion; then sR? rn nS a 


PROPOSITION LVIII. 


298. To determine the velocity with which a Ball moves, which 
being shot against. a Ballistic Pendulum, causes it to vibrate 
through a given Angle. 
The Ballistic Pendulum is a heavy block 
- of wood mn, suspended vertically by astrong 
horizontal iron axis at s, to which it is con- 
nected by a firmiron stem. This problem 
is the application of the last proposition, or 
of prop. 54, and was invented by the very 
ingenious Mr. Robins, to determine the ini- 
tial velocities of military projectiles ; a cir- 
cumstance very useful in that science ; and 
‘itis the best method yet known for deter- 
mining them wth any degree of accuracy. 





Let 


200 OR Oe i eres: 


Let c, R, o be the centres of gravity, gyration, and oscil- 
lation, as determined by the foregoing propositions : and let r 
be the point where the ball strikes the face of the pendulum ; 
the momentum of which, or the product of its weight and 
velocity, is expressed by the force f, acting ipo 
at ep, in the foregoing propositions. Now, 
Put p = the whole weight of the pendul. - 
.. 6 = the weight of the ball, — - Ree", 


g = se the dist. of the cen. of grav. 

o = so the dist. of the cen. of oscilla, 
r = sr=,/go the dist of cen. of gyr.. 

7 = sp the dist of the point of impact. 
» = the velocity of the ball, 





u = that of the point of impact r. 
c = chord of the arc described by o. 


By prop. 56, if the mass p be placed all at x, the pen- 

dulum will receive the same motion ree the blow in the 
2 

point p: andas sp? ; sR? ::p: —e p orp or =~ = Pp (prop. 

54), the mass which being placed at p, the me will still 

receive the same motion as” ‘before. Here then are two 


quantities of matter, namely, 6 and ap, the former moving 
; (a2 
with the velocity v, and striking the latter at rest ; to deter- 
mine their common velocity u, with which they will jointly 
proceed forward together after the stroke. In which case, 
2 the law of the impact of non-elastic bodies we have 
ba + Sop 
bee 

an of the ball in terms of w, the velocity of the point p, and 
the known dimensions and weights of the bodtes. 


u the velo- 





pr b:6::v: u, and therefore v = 


But now to determine the value of u, we must have re- 
course to the angle through which the pendulum vibrates ; 
for when the pendalum descends down again to the vertical 
position, it will have acquired the same velocity with which 

it began to ascend, and, by the laws of falling bodies, the 
welocity of the centre of oscillation is such, as a heavy body 
would acquire by freely falling through the versed sine of 
the arc described by the same centre 0. But the chord of 
that arc is c, and, its radius is 9; and by the nature of the 
‘errele, the chord is a mean Proportional between the versed 


CC 
sine and diameter, therefore 20: ¢::¢ are , the versed sine 
i 0 


of the are described by o. .Then, “ the laws of falling bodies 
, VJ 16 


? 


HYDROSTATICS, ae 


4/164; : Ven 5232) : vie. the velocity acquired by the ! 


point o in deadening dirs, the arc whose chord is c, where 


Qa. ¢ 
o=16), feet: and therefore o;:2::;¢ at oe whichis 


the velocity wu, of the point 8. 
Then, by substituting this value for u, the rslasity of the 


ball before found, becomes v aes c ees So that the 


velocity of the ball is directly as.the chord of the arc dee 
scribed by the pendulum in its vibration, 


SCHOLIUM. 


299. In the foregoing solution, the change in the centre 
of oscillation is omitted, which is caused by the ball lodging 
in the point p. But the ‘allowance for that small,change, and 
that of some other small quantities, may be seen in my Tracts, 
“(oat all the circumstances of this method are treated at full 

ength 

300. For an example in numbers of this method, suppose 
the weights and dimensions to be as follow : namely, 
p=570lb, — |Then 

= 1802. Ipdr.|biit-gop | _1°131x 94-39 +78} x845x570 


? 





= AU baa 1-131X94,3,X84¢ 
g = inc 18°73 
o=84zinc. |X =656-56, 
= 7°065 feet 
eH _ Qa 32p 5 108 
943. inc, And fo = S706 “065 = V 49:39 =n2: 1337, 
ec = 18: 73 inc. 


Therefore 656-56 X 2:1337 or 1401 feet, is the velocity, per 
second, with which the ball moved when it struck the pen- 
papas. 


OF HYDROSTATICS. 


301. Hyprosrarics is the science which treats of the press 
sure, or weight, and equilibrium of water and other fluids; 
Pepesially those that are non-elastic, 


302. A fluid is elastic, when it can be reduced into a less 
volume by compression, and which restores itself to its former 
bulk again when the pressure is removed’; as air. And it is 
non-elastic, when it is not compressible by such force; as 
water, &c. . 

Vou. Il. 27 PRO- 


202 HYDROSTATICS. 


er cr as ie _ PROPOSITION LIX. - 


303. i any Part of a Fluid be raised higher than the rest, abe 
any Force, and then left to itself; the higher Parts will de- 
 scend to the lower Places, and the Fluid will not rest, ull its 

Surface be quite even and level. 


For, the parts of a fluid being Giaty moveable every way, 

the higher parts will descend by their superior gravity, and 
_Yaise the lower parts, till the whole come to rest in a level or 
horizontal plane. 


304. Corol. 1. Hence, water that com- 
municates with other water, by means of 
a close canal or pipe, will stand at the 
same height in both places. Like as wa- 
ter in the two legs of a syphon. 


305. Corol. 2. For the same reason, if 
a fluid gravitate towards a centre ; it will 
dispose itself into a spherical figure, the 
centre of which is the centre of force. 
Like the sea in respect of the earth. 





PROPOSITION LX. 


306. When a Fluid is at Rest in a Vessel, the Base of which is 
Parallel to the Horizon; Equal Parts of the Base are Bouts 
Pressed by the Fluid. 


For, on every equal part of this base there is an equal 
column of the fluid supported by it. And as all the columns 
are of equal height, by the last proposition they are of equal 
weight, and therefore they press the base equally ; that is, 
equal parts of the base sustain an equal pressure. — 


“ ; 
307. Corol. 1. All parts of the fluid press equally at the 

same depth. For, if a plane parallel to the horizon be con- 

ceived to be drawn at that depth: then the pressure being 

the same in any part of that plane, by the proposition, there- 

fore the parts of the fluid, instead of the plane, sustain the 
same pressure at the same depth. 


308. Corol. 2. The pressure of the fluid at any dene is 
as the depth of the fluid. For the pressure is as the weighty. 
andthe weight is as the height of the fluid. 

309. toils 


PRESSURE OF FLUIDS. 203. 


309. Corol. 5. The pressure of the fluid on any horizon-’ 
tal surface or plane, is equal to the weight of a column of the 
fluid, whose base is equal to that plane, and altitude is its depth 
below the upper surface of the fluid. 


PROPOSITION LXI. 


310. When a Fluid is Pressed by its own Weight, or by any 
other Force; at any Point it Presses Equally, in all Direc- 
tions whatever. Sah os : | 


‘Tuts arises from the nature of fluidity, by which it yields 
to any force in any direction. If it cannot recede from any 
force. applied, it will press against other parts of the fluid in 
the direction of that force. And the pressure in all direc- 
tions will be the same: for if it were less in any part, the 
fluid would move that way, till the pressure be equal every 
Way. 


311. Corol. 1. In a vessel containing a fluid ; the pressure 
is the same against the bottom, as against the sides, or even 
upwards at the same depth. 


312. Corol. 2. Hence, and from 3 
the last proposition, if ascp bea i Aen 5 
vessel of water, and there be taken, = | 
in the base produced, pe, to repre- Se 
sent the pressure at the bottom ; 
joining ab, and drawing any paral- 
lels to the base, as Fe, H1; then LZ = 
shall re represent the pressureat E OD —¢ 
the depth ac, and u1 the pressure : 
at the depth a1, and so on ; because the parallels ra, 1, ep, 
by sim. triangles are as the depths - - - - ac, at, ap: 
which are as the pressures, by the proposition. 

And hence the sum of all the re, n1, &c. or area of the 
triangle ape, is as the pressure against all the points c, 1, &c. 
that is, against the line ap. But as every point in the line cp 
is pressed with a force aS DE, and that thence the pressure on, 
the whole line cp is as the rectangle Ep . pc, while that against 
the side is asthe triangle apg or ap. pE; therefore the pres- 
sure on the horizontal line pc, is to the pressure against the 
vertical line pa, as pc to pa. And hence, if the vessel be 
an upright rectangular one, the pressure on the bottom, or 
whole weight of the fluid, is to the pressure against one side, 
as the base is to half that side. Therefore the weight of the 
fluid is to the eee against all the four upright aaa 

é tne 





y 


the base is to half the upright surface. And the same holds 
true also in any upright vessel, whatever the sides be, or ina 
cylindrical vessel. .Or in the cylinder; the weight of the 
fluid; is to the pressure against the upright surface, as the ra- 
dius of the base is to double the altitude. 


Also, when the rectangular prism becomes a cube, it ap- 
pears that the weight of the fluid on the base, is double the 
pressure against one of the upright sides, or half the pres- 
gure against the whole upright surface. 


313. Carol. 3. The pressure of a fluid against any upright 
surface, as the gate of a’sluice or canal, is equal to half the 
weight of a column of the fluid whose base is equal to the 
surface pressed, and its altitude the same as the altitude of 
that surface. For the pressure ona horizontal base equal to 
the upright surface, is equal to that column ; and the pres- 
sure on the upright surface, is but half that on the base, of 
the same area. 


So that, if 6 denote the breadth, and d the depth of such a 
gate or upright surface ; then the pressure against it, is equal 
to the weight of the fluid whose magnitude is 1bd2 =1aB , ap?. 
Hence, if the fluid be water, a cubic foot of which weighs 
1000 ounces, or 621 pounds ; andif the depth ap be 12 feet, 
the breadth az 20 feet ; then the content; or 14B . ap?, is 1440 
feet ; and the pressure is 1440000 ounces, or 90000 pounds, 
or 40, tons. 


PROPOSITION LXII. 


314.. The pressure of a Fluid on a Surface any how immersed 

init, either Perpendicular, or Horizontal, or Oblique; vs 
Equal to the Weight of a Column of the Fluid, whose Base is 
equal to the Surface pressed, and its Altitude equal to the 

| Depth of the Centre of Gravity of the Surface pressed below 
the Top or Surface of the Flucd. - 


For, conceive the surface pressed to be divided into innu- 
merable sections parallel to the horizon; and let s denote 
any one of those horizontal sections. also d its distance or 
depth below the top surface of the fluid. Then, by art. 309, 
the pressure of the fluid on the section is equal to the weight 
of ds ; consequently the total pressure on the whole surface 
is equal to all the weights ds. But,if 6 denote the whole 
surface pressed, and g the depth of its centre of gravity be- 
_ tow the top of the fluid; then, by art. 256 or 259, bg is equal 
: to 


PRESSURE OF FLUIDS. 208 


to the sum of all the ds. Consequently the whole pressure 
of the fluid on the body or surface 6 is equal to the weight of 
the bulk bg of the fluid, that is, of the column whose base is 
the given surface 6, and its height is g the depth of the centre 
of gravity in the fluid. 


PROPOSITION LXIII. 


315. The Pressure of a Fluid, on the Base of the Vessel in which 
at is contained, is as the Base and Perpendicular Altitude : 
whatever be the Figure of the Vessel that contains tt. 


Ir the sides of the base be upright, so that db 
it be a prism of a uniform width throughout ; | 
then the case is evident; for then the base 
supports the whole fluid, and the pressure 1s 
just equal to the weight of the fluid. 

But if the vessel be wider at top than bot- 
tom; then the bottom sustains or is pressed 
by, only the part contained within the up- 
right lines ac, bo; because the parts aca, 
ppb are supported by the sides ac, sp; and 
those parts have no other effect on the part 
aboc than keeping it in its position, by the la- 
_teral pressure against ac and bp, which does 
not alter its perpendicular pressure downwards. And thus 
the pressure on the bottom is less than the weight of the con- 
tained fluid. = 

And if the vessel. be widest at bottom; then 
the bottom is still pressed with a weight which 
ig equal to that of the whole upright column 
abpc. For, as the parts of the fluid are in 
equilibrio, all the parts have an equal pressure 
at the same depth; so that the parts within cc’ 
and dp press equally as those in cd, and there- 
fore equally the same as if the sides of the vessel had gone 
upright to a_and b, the defect of fluid in the parts aca and 
sob being exactly compensated by the downward pressure 
_ or resistance of the sides ac and wp against the contiguous 

fluid. And thus the pressure on the base may be made to | 

exceed the weight of the contained fluid, in any proportion 

- whatever.” 5 

So that, in general, be the vessels of any figure whatever, 
regular or irregular, upright or sloping, or variously wide 
and narrow in different parts, if the bases and perpendicular 
altitudes be but equal, the bases always sustain the same 
pressure. And as that pressure, in the regular upright 
| vessel, 








206 _ HYDROSTATICS, 


vessel, is the whole column of the fluid, which is as the base 
and altitude ; ; therefore the pressure in all figures. is in that. 
same ratio. 
316. Corol. 1. Hence, when the heights are enna the: 
pressures are as the bases. And when the bases are equal, 
the pressure is as the height. But when both the heights and 
bases are equal, the pressures are equal in all, ehoden their 
contents be ever so different, 
~ 317. Corol. 2. The pressure on the fake of any vessel 18 
‘the same as on that of a cylinder, of an equal base and height. 
318. Corol. 3. If there be an inverted sy- 
phon, or bent tube, asc, containing two dif- 
ferent fluids cp, asp, that balance each other. 
or rest in equilibrio ; then their heights in 
the two legs, ar, cp, above the point of meet- 
ing will be reciprocally as their densities. 
-For, if they do not meet at. the bottom, A 
the part Bp balances the part Be ; and there- 
- fore the part cp balances the part ar; that is, EK) 
the weight of cp is equal to the weight of ar. 
And as the surface at p is the same where 
they act against each other, therefore ar: cp 
:: density of cp : density of aE. | 
So, if cp be water, and ae quicksilver, which is near 14 
times heavier; then cp will be = 14ar; that ts, if ae be 1 
inch, cp will be 14 inches ; if az be 2 inches, cp will be 28 
inches ; ; and so on. 


Be 





La) 





ANGOLA UMA 


€ 


PROPOSITION LXI1V. 


319. If a Body be Iinmersed in a Fluid of the same Density or 
Specific Gravity; at will Rest in any Place where it ts put. 
But @ Body of Greater Density will Sink ; and one of a Less 
Density will Rise to the Top, and Float. ~ 


Tue body, being of the same den- 
sity or of the same weight with the © 
like bulk of the fluid, will press the 
fluid under it, just as much as if its 
space was filled with the fluid itself. 
The pressure then all around it will 
be the same as if the fluid were in 
its place ; consequently there 1s no 
force, neither upward nor down- 
ward, to put the body out of its place. 
And therefore it will remain where- 
ever itis put. _ 





PRESSURE OF FLUIDS. 207 


But if the. poey be lighter ; its pressure downward will be 
less than before, and less than the water upward at the same 
depth ; therefore the great force will overcome the less and 
push the body Upward to a. 


And if the body be heavier than the fluid, the pressure 
downward will be greater than the fluid at the same depth ; 
therefore the greater ere will Collar “and Maes she body 
down to the bottom at c. 


320. Corol. 1. A body immersed in:a fluid, loses as ‘much 
weight, as an equal bulk-of the fluid weighs. And the fluid 
gains the same weight. Thus, if the body be of equal densi- 
ty with the fluid, it loses all its weight, and so requires no 
force but the fluid to sustain it. If it be heavier, its weight 
in the water will be only the difference between its own weight 
and the weight of the same bulk of water; and it requires a 
force to sustain it just. equal to that difference. But if it be 
lighter, it requires a force equal to the same difference of 

‘weights to keep it from rising up in the fluid. 


321. Corol.2. The weights lost, by immerging the same bo- 
dy in different fluids, are as the specific gravities of the fluids. 
And bodies of equal weight, but different bulks, lose in the 
same fluid, weights which are reciprocally as the specific 
gravities of bodies, or directly as their bulks, 


322. Corol. 3. The whole weight of a body which’ will 
float in a fluid, is equal to"as much of the fluid, as the im- 
mersed part of the body takes up, when it floats. F or the 
pressure under the floating body, is just the same as so much 
of the fluid as is equal to ee immersed part; and therefore 


the weights are the same. 


$23. Corol. 4. Hence the magnitude of the whole body, 
is to the magnitude of the part immersed, as the specific gra- 
vity of the fluid, is to that of the body. For, in bodies of 
equal weight, the densities, or specific gravities, are ae 
ony as their magnitudes. 


$24. Corol. 5. And because when the mela of a body 
taken in a fluid, is subtracted from its weight out of the fluid, 
the difference is the weight of an equal bulk of the fluid ; this 
therefore is to its weight in the air, as the specific eravity of 
the fluid, is to that of body. 
Therefore, if w be the weight of a body in air, 
--- @ its weight in water, or any fluid, 
s the specific gravity of the body, ‘and 
s the specific gravity of the fluid ; : 
. then 


208  HYDROSTATICS, 


then w—w : w:: 8:8; which proportion will give either of 
those pia Cian the one from the other. 


Thus s== 
w— 





“—s, the specific gravity of the body ; 


—"s, the specific gravity of the fluid. 





Ww 
and s== 


So that the specific gravities of bodies, are as their weights 
in the air directly, and their loss in the same fluid inversely, 


$25. Corol. 6. And hence, for two bodies connected to- 
gether, or mixed together into one compound, of different 
specific grayities, we have the following equations, denoting 
their weighis and specific gravities, as below, viz. 


H = weight of the heavier body in air, 

h = weight of the same in water, 

u = weight of the lighter body in air, 

3 = weight of the same in water, 

= weight of the compound in air, 

¢ = weight of the same in water, 

w@ = the ‘specific gravity of water. Then, 

Ist, (Hh) s=H, From, which equations may be found 

2d, (n—1) s=Lw, | any of the above qanetiice, in terms of 

3d, (c—c) f=cw, | the rest. 

4thha-+Li=c, Thus, from one of the first three 

ey h+-l=c, equations, is found oe specific gravity 
3 bc 

} page ay al =F 
spookvie icient of the body by its loss in aad and saylpplie 
ing by the specific gravity of water. 


But if the body 1 be lighter than water; then / will be 
negative, and we must divide by t-++/ instead of 1-1, -and to 
find / we must have recourse to the compound mass c; and 


because, from the 4th and 5th equations, r—l=c—c—u—h, 
__“" _____. that is, diyide the absolute 
c—c)—(H—h 
weizht of the light body, by the difference between the losses 
in water, of the compound and heavier Bony and multiply by 


: uf” 
from the last equation, 

Also, if it were required to find the quantities of two 3 ingre- 
dients mixed in a compound, the 4th and 6th equations would 
give their values as follows, viz. 


s its spec. gravity ; 
s its spec. gravity ; 


f its spec. gravity ; 








| of any body, as guetay De ay dividing the 





therefore s= 


as found 





the specific gravity of water. Or thus, s=- a cabs 


= 


SPECIFIC GRAVITY. 209 
~2(-f)s.. 


ea, s—s) f ; . NGC o> s) f 4 “ 
the quantities of the two ingredients 4 and 1, in the com- 
poundec. And so forany other demand: 





PROPOSITION LXV. 
To find the Specific Gravity of a Body. 


326. Cast 1.—When the body is heavier than water : weigh 
it both in water and out of water, and take the differénce, 
which will be the weight lost in water. Then, by corol. 6, 
Bw 
B—b 
water, 6 its weight in water, s its specific gravity, and w the 
specific gravity of water. That is, : : 

. As the weight lest in water; 
Is to the whole or absolute weight, 
So is the specific gravity of water, 
To the specific gravity of the body: 


prop. 64, s= , where 8 is the weight of the body out of 





Exampze. If a piece of stone weigh 10Ib, but in water 
only 62 lb, required its specific gravity, that of water being 
1000 ?. (oe | Ans. 3077. 


327. Case 1.—When the body is lighter than water, so that 
at will not sink: annex to it a piece of another body, heavier 
than water, so that the mass compounded of the two may sink 
together. Weigh the-denser body and the compound mass, 
separately, both in water and out of it; then find how much 
fach loses in water, by subtracting its weight in water from 
its weight in air ; and subtract the less of these remainders 
from the greater. Then say, by proportion, 

As the last remainder, — | 

Is to the weight of the light body in air, — 
So is the specific gravity of water, 

To the specific gravity of the body. 


rary ° ° >, NY ane geo » LW sin - 
_ That is, the specific gravity is s= (exe tt asi (hy 2 


by cor. 6, prop. 64. . 
Examrre. Suppose a piece of elm weighs 15!b in air ; 
and that a piece of copper, which weighs 1!8lb in air and 
161b in water, is affixed to it, and that the compound weighs 
6tb water ; required the specific gravity of the elm ? 
Ans. 600, 
Vou. Ii, 28. 328. Case 


210 HYDROSTATICS. « 


328. Case m1.—For a fluid of any sort.—Take a piece of 
a body of known specific gravity; weigh it both in and out 
of the fluid, finding the loss of | via by tahing the differ- 
ence of the two; then say, : 


As the whole or absolute weight, 

Is to the loss of weight, 

So is the specific gravity of the solid. 
To the apsriic gravity of the fluid. 


That is, the spec. grav. tr Oy by cor. 6, pr. 64: 


. Exameie. A piece of cast iron Saighes 35,51, ounces in 
4 fluid, and 40 ounces out of it; of what specific gravity is 


that fluid? ? Ans. 1000. 


PROPOSITION LXVI. 


329. To find the Quantities of Two hei in a Given 
Compound. 


Take the three differences. of every pair of the three spe- 
cific gravities, namely, the specific gravities of the compound 
and each ingredient; and multiply each specific gravity by 
the ditference of the other two. Then say, by peeperiel, 


As the greatest product, 

Is to the whole weight of the compound, 
-$o is each of the other two products, 
Fo the weights of the two ingredients. 


ek (fs) 8 _(s—f)s 
That | is, H = c = the one, andi = __c, the 
(s—s)\f (s—s)f’ 
other, by cor. 6, prop. 64. ae Weg 





Examp.e. A composition of 1121b being made ‘of tin and. 
copper, whose specific gravity is found to be 8784; required 
the quantity of each ingredient, the specific gravity of tin being 
7320, and that of copper 9000 ? 

Answer, there is J00Ib of copper, mM Te 
and consequently 12lb of tin, i LAS em RpORtOn. 


SCHOLIUM, 


2 333. The specific gravities of several sorts of matter as 
found from experiments, are expressed by the numbers an- 
nexed to their names in the following Table : 


A Table 


ej 


SPECIFIC GRAVITY. 212 


A Table is Specific Gravities Oe Bodies. 


Platina (pure) ~ - 23000\Clay = = + ++. + 2160 
Fine gold - - - - 19400/Brick. - -. = © = = .= 2000 
Standard gold- - - -17724\Commonearth - - - 1984 
Guicker pieey  J14000\Nitre - - - - - = 1900 
Sia res common) - 13600)[vory - + - - - = 1825 
Lead --- = - - -.11325\Brimstone. - - - - 1810 


Fine silver - + = + 11091|/Solidgunpowder-. -  - 1745 
Standard Silver - 10535|Sand - +. - -. + = 1520 


Copper - -, - - +  9000/\Coal .- +--+ =. + + 1950 
Copper halfpence - - 8915|Box-wood- - - - - 1030 
Gun metal - - -+- 8784\Sea-water- - - - = 1030 
Cast brass - - - - 8000/\Common-water - = - 1000 
Steel - .- - -.- - 78500ak - - - = 925 
iron - - - - + + 7645|Gunpowder, gloss shaken 9357 
CastIron - - - +. 7428|Ditto, inaloose heap - 836 
Tin -- --- - +. 7320)Ash - - - - = - 800 
Clear crystal glass- - .3150/Maple- - - - - - 755 
Granite: 20-0. i+. “=e = GOOD Elm 4+ - She Ss 600 
Marble and hard stone 2700/Fir- - - - = - + 5850 
Common green glass - 2600;\Charcoal - - - - - 

Flint - ~  - - -) = 2570\Cork - - - = 2 «+. 240 
Commen stone -. - -  2520\Air ata mean state - -  .12 


331. Note. The several sorts of wood are supposed to be. 
dry. Also, as a cubic foot of water weighs just 1000 ounces 
avoirdupois, the numbers in this table express not only the 
specific gravities of the several bodies, but also the weight of 
a cubic foot of each, in avoirdupois ounces; and therefore, 
by proportion, the weight of any other quantity, or the quan- 
tity of any other weight, may be known, as in the next two 
propositions. 

_ PROPOSITION LXVILI. ° 


382. To find. the Magnitude of any Body, from its Weight, 


_ As the tabular specific gravity of the body, 
Is to its weight in avoirdupois ounces, 
So is one éubic foot, or 1728 cubic inches, 
To its content in feet, or inches, respectively. 
5 Riterninle 1. Reguired the content of an irregular block of 
common stone, which weighs 1 cwt. or 112lb? 
| : Ans. 12282216 cubic inches. 
Example 2 eaiote many cubic inches of gunpowder are there 
in Ib weight ? ? Ans. 291 cubic inches nearly. 
Example 3. How way sine feet are there in a ton weight 
of dry oak ? Ans, 38132 cubic feet. 
* PROPOSITION 


812 HYDRAULICS. 
PROPOSITION LXVIII. 


a 333. To jind the Weight of a Body from its Magnitude. 


_As one cubic foot, or 1'728 cubic inches, : 
~ Is to the content of the body, — 

So is the tabular specific gravity, | 

To the weight of the body. 

Example 1. Required the weight of a block of marble, 
Whose length is 63 feet, and breadth and thickness each 12 
‘feet; being the dimensions of one’ of the stones in the walls 
‘of Balbeck ? 

Ans. 6834, ton, which ts nearly equal to the burden of 
| an East- {naj ship. 

Example 2. What is the weight of 1 pint, ale measure, of 
gonpowder.’ ? Ans. 19 oz. nearly. 

Example 3. What is the. weight of a block of dry oak, 
which measures 10 feet in length, 3 feet broad, and 21 feet deep 
or thick ? Ans. "4335141b. 


OF HYDRAULICS, 


334. Hypravtics i is the science which treats of the motion 
of fluids, and the forces with which they act upon bodies. 


PROPOSITION LXIX. 


B35. yf a Fluid Run through a Canal or River, or Pipe of va 
rious Widths, always filling it: the Velocity of the Flud in 
different Parts of it 4B, cp, will be reciprocally as the Trans- 
verse Sections in those Parts. 
Tnar tis, veloc: ata: veloc. 

atc :: cD: AaB; where as and 

cp denote, not the diameters 
at a and 8, but the areas or 
sections there. 

For, as the channel is always equally fall, the quantity of 
water running through ap is equal to the quantity running 
through cp, in the same time ; that is the column through 
AB is equal to the column through cp, in the same time ; or 
ap X length of its column =cp X length of its column ; there- 
fore an: cp :: length of column through cn: length of co- 
jumn through as. But the uniform velocity of the water, is 
as the space run over, or length of the columns; therefore 
ab.tcp:: velocity through cp: velocity through ap. _ 

336. Corol. Hence, by observing the velocity at any place 
ap, the quantity of water discharged ina second, or any other 

time, 





SPOUTING OF FLUIDS. 213 


time, will be found, namely, by multiplying the section az by 
the velocity there. 

But if the channel be not a close pipe or tunnel, kept al- 
ways full, but an open canal or river; then the velocity j in 
all parts of the section will not be the same, because the 
velocity towards the bottom and sides will be diminished by 
the friction against the bed or channel, and therefore a me- 
dium among the three ought to be taken. So, if the pee 


at the top be - 100 feet per minute, 
that at the bottom - 60 
and that at the sides. - ie 

3) 210 sum: 


dividing their sum by 3, gives 70 for the mean velocity, which 
is to be multiplid by the section, to give ths shay discharg- 
ed in a minute. 


PROPOSITION LXX. : 


337. The Velocity with which a Fluid Runs out by a Hole on 
the Bottom or Side of a Vessel, is Equal to. that which 1s Ge- 
nerated by Gravity through the Height of the Water above the 
Hole ; that is, the Velocity of a Heavy Body acquired by Fal!- 
ing freely through the Herght as. 


Divipe the altitude ae into a great 
number of very small parts, each being 
1, their number a, or a = the altitude 
ABy 
Now, by prop. 61, the pressure of 
the fluid against the whole 8, by which 
the motion is generated, is equal to the 
weight of the column of fluid above it, 
that is, the column whose height is as or a, and base the area 
of the holes. Therefore the pressure on the hole, or small 
part of the fluid 1, is to its weight, or the natural force of 
gravity,asato 1. But, by art, 28, the velocities generated 
in the same body in any time, are as those forces ; and be- 
cause gravity generates the pion 2 in descending. through 
the small space 1, therefore 1:a:: 2: 2a, the velocity ge- 
nerated by the pressure of the eatuihe ‘of fluid in the same 
time. But 2a is also, by corol. 1, prop. 6, the velocity ge- 
nerated by gravity in descending ‘through aorap. ‘Lhat is, 
the velocity of the issuing water, is equal to that which is ac- 
quired by a body in fing through the height az: i 

te 





214 -. HYDRAULICS: 


The same otherwise. 


"Because the momenta, or quantities of motion n pen deatel itl 
two given bodies, by the same force, acting during the same 
or an equal time, are equal. And as the force in this ease, is 
‘the weight of the superincumbent column of the fluid over 
the hole. Let the one body to be moved, be that column it- 
self, expressed by ah, where a denotes the altitude aB, and kh 
the area of the hole ; and the other body is the column of the 
fluid that runs out uniformly i in one second suppose, with the 
middle or medium velocity of that interval of time, which is 

tho, if v be the whole velocity required. Then the mass thv, 
with the velocity v, gives the quantity of motion thu X v or 
tho? , poner eed) is one second, in the spouting water : also 2g, 
or 32} feet, is the velocity generated in the mass ah during 
the same interval of one ‘second ; ; consequently ah * 2g, or 
Zehg, is the motion generated in the column ah in the same 
time of one second. But as these two momenta must be equal, 
this gives Iho? ==2ahg: hence then v? = 4ag, and v = 2,/ag, 
for the value of the velocity sought : which therefore is ex- ° 
actly the same as the velocity generated by the gravity in fall- 
“ing throngh the space a, or the whole height of the fluid.* 
‘For example, if the fluid were air, of the whole height of 
the atmosphere, sayy A uniform, which is about 5} miles, 
or 27720 feet =a. Then 2,/ag = 2,/27720 X 16-25 "= 1335 
feet =v the velocity, that i is, the velocity with which common 
air would rush into a vacuum. 
|. 338. Corol. 1. The velocity, and quantity run out, at dit: 
ferent depths, are as the square roots of the depths. For the 
velocity acquired in falling through an, is as 4/AB. 

339. Corol. 2. The fluid spouts out with the same’ veloci- 
ty, whether it be downward or upward, or sideways ; because 
the pressure of fluids is the same in all directions, at the 
same depth. And therefore, if an adjutage be turned up- 

_ ward, the jet will ascend to the height of the surface of the 
water in the vessel: And this is confirmed by experience, 
by which it is found: that jets really. eee nearly to the 


a H { >= ‘ 
a RT ERTS 


* In this investigation the author nses the whole momentum ah X 2g which is 
wencrated | in one second by the ‘gravity of the mass ah; but he does not use the 

4 whole momentum hv & v which is also generated by the same force in the sarrie 
time, instead of which he uses only half the latter momentum ; on this account 
hhis solution appears to me to be erroneous. The two momenta ah X 2g and hy 
% v produced i in one second by the. same force ought to be equal, which gives ve 


erie instead of the equation »~ * dag as found by Dr. Hutton. * Ed. 


height 


SPOUTING OF FLUIDS. ath 


height of the reservoir, abating a small quantity only, for the 
friction against the sides, and some resistance from the air 
and from the oblique motion ef the fluid in the hole. 

340. Corol. 3. The quantity run vut in any time, is equal 
to a column or prisin, whose base is the area of the hole, and 
its length the space described in that time by the velocity ac- 
quired by falling through the altitude of the fluid. And the 
quantity 19 the same, whatever be the figure of the orifice, if 
it is of the same area. | 7 

Therefore, if a denote the altitude of the fluid, 

\: and & the area of the orifice, 

also g==167, feet, or 193 inches ; d 
then 2h,/ag will be the quantity of water discharged in a 
second of time; or nearly 87;4,/e cubic feet, when a and & 
are taken in feet. TNS Paleo 

So, for example, if the height a be 25 inches, and the ort- 
fice 4=1 square inch; then 2h,/ag=2,/25 X 193139 cu- 
bic inches, which is the quantity that would be discharged per 
second, 3 

SCHOLIUM. 


. 341. When the orifice is in the side of the vessel, then the 
welocity is different in the different parts of the hole; being 
less in the upper parts of it than in the lower. However,’ 
when the hole is but small, the difference is inconsiderable, 
and the altitude may be estimated from the centre of the hole 
to obtain the mean velocity. But when the orifice is pretty 
large, then the mean velocity is to be more accurately com- 
puted by other principles, given in the next proposition. — . 
342. It is not to be expected that-experimenis, as to the 
quantity of water run out, will exactly agree with this theory, 
both on account of the resistance of the air, the resistance of 
the water against the sides of the orifice, and the oblique mo- 
tion of the particles of the water in entering it: For, it ts 
not merely the particles situated immediately in the column 
over the hole, which enter it and issue forth, as if that colamn 
only were in motion; but also particles from all the surround- 
ing parts of the fluid, which is in a commotion quite around ; 
and the particles thus entering the hole in ali directions, strike 
against each other, and impede one another’s motion: from 
which it happens, that it is the particles in the centre of the 
hole only that issue out with the whole velocity due to the 
entire height of the fluid, while the other particles towards 
the sides of the orifices pass out with decreased velocities : 
and hence the medium velocity through the orifice, is some- 
what less than that of a single body only, urged with the same 
pressure of the superincumbent column of the fluid. And 
experiments 


216 - HYDRAULICS. 


experiments on the quantity of water discharged through 

apertures, show that the quantity must be diminished, by those 

causes, rather more than the fourth. part, when the orifice is_ 

~ small, or such as to make the mean velocity ‘nearly equal to 
that in a body falling through 3 the hpight. of the fluid above ' 
the orifice. oe 


343. Experiments have also been made on the extent to 
which the spout of water ranges on a horizontal plane, and 
compared with the theory, by calculating it. as a projectile 
discharged with the velocity acquired by descending through 
the height of the. fluid. For, when the aperture is in the 
side of the vessel, the fiuid spouts out horizontally with a uni- 
form velocity, which, combined with the perpendicular velo 
city from the action of gravity, causes the jet to form the curve 
of a parabola. Then the dis- | 
tances to which the jet will spout. 
on the horizontal plane se, will 
be as the roots of the rectangles 
of the segments ac . cB, ap . DB, 
AE. EB. For the spaces Br, BG, 
are as the times and horizontal. 
velocities ; but the velocity is as 
4/ac; and the time of the fall 
which is the same as the time 
of moving, is as ,/ cB; therefore the distance. BF is ag 


fac . cB; and the tiauee BG as4/ Ap.ps. And hence, if 
two holes are made equidistant from the top and bottom, they 
will project the. water to the same distance; for if ac=kx, 
then the rectangle ac . cB is equal the rectangle AE . EB! 
which makes er the same for both. Or, if on the diameter 
aB a semicircle be described; then, because, the squares of 
the ordinates cH, pI, BK are equal to the rectangles ac . rx, 
&c.; therefore the distances Br, Bc are as the ordinates cu, 
pr. And hence also it follows, that the projection from the 
middle point p will be farthest, for pi is the greatest ordinate. 





- These are the proportions of the distances: but for the 
Baclive distances, it will be thus. The velocity through 
~ any hole c, is such as will carry the water horizontally 
ee a space equal to 2ac in the time of falling through 

: but, after quitting the hole, it describes a parabola, and 
hiss to F in. the time a.body will fall through cp; and 
to find this distance, since the times are as the roots of 


the sgidiatthebelops yLdOi2 GfCB.>: + Bef «08 cB == 


PNEUMATICS. 217 


2cH = er, the space ranged on the horizontal plane. And the 
greatest range BG = 2p1, or 2ap, or equal to aB. 

And as these ranges answer very exactly to the experiments, 
this confirms the theory, as to the velocity assigned. 


PROPOSITION LXXI. 


344. If a Notch or Slit en in form of a Parallelogram, be cut 
am the Side of a Vessel, Full of Water, an; the Quantity of 
Water flowing through tt, will be 2 of the Quantity flowing 
through an equal Orifice, placed at the whole Depth rc, or 
at the Base au, in the Same Time ; 1t being supposed that the 
Vessel is always kept full. 


For the velocity at cu is to the velo- 
city at IL, as 4/ EG to 4/ EI; that is, as 
GH or 11 to 1k, the ordinate of a para- 
bola exs, whose axisisee. Therefore 
the sum of the velocities at all the points 
1, 1s to as many times the velocity at c, 
as the sum of all the ordinates 1x, to the 7 
sum of allthe 11’s; namely, as the area 
of the parabola ecu, is to the area Ecur ; that is, the quanti- 
ty running through the notch eu, is to the quantity running 
through an equal horizontal area placed at cu, as EGHKE, to 
EGHF, or as 2 to 3; the area of a parabola being 2 of its cir- 
cumscribing parallelogram. | 

. 345. Corol. 1. The mean velocity of the water in the notch, 
is equal to 2 of that at cu. | 

346. Corol. 2. The quantity flowing through the hole ienz, 
is to that which would flow through an equal orifice placed as 
low as GH, as the parabolic area Iank, 1s to the rectangle 1cur., 
As appears from the demonstration. : 


==6 





D 


aaa 


OF PNEUMATICS. 


347. Pneumatics is the science which treats of the pro- 
perties of air, or elastic fluids. 


PROPOSITION LXXIL 


348, Air isa Heavy Fluid Body; and it surrounds the Earth, 
and Gravitates on all Parts of its Surface. 


That it is a fluid, is evident from its easily yielding to | 


Tuese properties of air are proved by iain 
y 
Von, I. 29 @ 


218 . PNEUMATICS. 


the feast force impressed on it, without ee a sensible re- 
sistance. 

But when it is moved briskly, by any means, as by a bn or 
a pair of bellows; or when any body is moved very briskly 
through it ; in these cases we become sensible of it as a body, 
by the resistance it makes in such motions, and also by its im- 
pelling or blowing away any light substances. So that, being . 
capable of resisting, or meving other bodies, by its impulse, it 
must itself be a body, and be heavy, like all other bodies in 
proportion to the matterit contains ; and therefore it will press 
on all bodies that are placed under it. 

Also, as it is a fluid, it spreads itself all over on the earth ; 
and, like other fluids, it gravitates and presses everywhere on 
the earth’s surface. 


349. The.gravity and pressure of the air 
is also evident from many experiments. 
Thus, for instance, if water, or quicksilver, 
be poured into the tube ace, and the air be 
suffered to press on it, in both ends of the 
tube, the fluid will rest at the same height 
in both legs: but if the air be drawn out of 
one end as E, by any means; then the air 
pressing on the other end a, will press down 
the fluid in this leg at B, and raise it up in the other to p, as 
much higher than at 8, as the pressure of the air is equal to. 
From which it appears, not only that the air does really press, 
but also how much the intensity of that pressure is equal to. 
And this is the principe of the barometer. 





iii cae) LXXIII. 


350. The Air is also an Elastic F fit; being Condensible and 
Expansible. And the Law it observes is this, that its Density 
and Elasticity are proportional to the Force or Weight which 
Compresses it. ) 


Tus property of the air is proved by many experiments. 
Thus, if the handle of a syringe be pushed inward, it will con- 
dense the inclosed air into less space, thereby showing its 
condensibility. But the included air, thus condensed, is felt to 
act strongly against the hand, resisting the force compressin 
it more and, more ; and on withdrawing the hand, the handle is 
pushed | back again to where it was at first. Which shows that 
the air is elastic. | sks dane 

i | (351. Again, 


ELASTICITY OF AIR. 919 


351. Again, fill a strong bottle half full of 
water; then insert a small glass tube into it, 
putting its lower end down near to the bot- 
tom, and cementing it. very close round the 
mouth of the bottle. Then, if air be strong- 
ly injected through the pipe, as by blowing 
with the mouth or otherwise, it will pass 
through the water from the lower end, as- 
cending into the parts before occupied with 
air at B, and the whole mass of air become 
there condensed, because the water is not 
compressible into a less space. But, on removing the force 
which injected the air a, the water will begin to rise from 
thence in a jet, being pushed up the pipe by the increased 
elasticity of the air 8, by which it presses on the surface of 
the water, and forces it through the pipe, till as much be ex- 
pelled as there was air forced in; when the. air at 8 will be 
reduced to the same density as at first, and the balance being 
restored, the jet will cease. 

352. Likewise, if into .a jar of water 
as, be inverted an empty glass tumbler 
cp, or such-like, the mouth downward ; 
the water will enter it, and partly fill it, 
but not near so high as the water in the 
jar, compressing and condensing the air 
into a less space in the upper parts c, and 
causing the glass to make a sensible. re- 
sistance to the hand in pushing it down. 
Then, on removing the hand, the elasticity of the internal 
condensed air throws the glass up again. All these showing 
that the air is condensible and elastic. 

353. Again, to show the rate or proportion A 
of the elasticity to the condensation: take a 
long crooked glass tube,equally wide through- 
out, or at least in the part Bp, and open at a, 
but close at the other ends. Pour in a little. 
quicksilver at a, just to cover the bottom to 
- the bend at cp, and to stop the communica- 
tion between the external air and the air in 
bp. Chen pour in more quicksilver, and 
mark the corresponding heights at which it 
stands in the two legs: so, when it rises to 
u in the open leg, ac, let it rise to & in the 
close one, reducing its included air from the 
natural bulk wp to the contracted space BE, 














ae 






iil 








22%) PNEUMATICS. 


by the pressure of the column ne ; and when the quicksilver 
stands at 1 and x, in the open leg, let it rise to r and c in the 
_ other, reducing the air to the respective spaces Er, Bc, by the 
weights of the columns rf, xg. Then it is always found; that 
the condensations and elasticities are as the compressing 
weights or columns of the quicksilver, and the atmosphere 
together. So, if the natural bulk of the air Sd be compress- 
ed into the spaces BE, BF, Bc, which are 3, 2,1, of up, or as 
the numbers 3, 2. 1: then the atmosphere, together with the 
corresponding columns ue, 1f, Kg, are _ are tobe in the 
same proportion reciprocally, viz. as1,1,1, or as the num- 
bers 2,3, 6. And then ne=1a, 1f = a, and kg= 3a; where 
A is the weight of atmosphere. Which show, that the con- 
densations are directly as the compressing forces. And the 
elasticities are in the same ratio, since the columns im ac are 
sustained by the elasticities in pp. 

From the foregoing principles may be deduced many useful 
remarks, as in the following corollaries, viz. 


354. Coro]. 1. The space which 
any quantity of air is confined in, 
is reciprocally as the force that 
compresses it. So, the forces which © 
confine a quantity of airin the cy- 
lindrical spaces Ac, BG, cc, are 
reciprocally as the same, or reci-. 
procally as the heights an, Bp, cp, 
And therefore if to the two perpendicular lines pa, DH, as 
asymptotes, the hyperbola rxt be described, and the ordinates 
AI, BK, CL be drawn ; then the forces which confine the air 





in the spaces AG, BG, CG, 
will be directly as the corresponding ordinates At, BE, CL, 
since these are reciprocally as the abscisses AD, BD, €D, 


by the nature of the hyperbola. 


355. Corol. 2. All the air near the earth, is in a state of 
compression, by the weight of the incumbent atmosphere. 


356. Corol. 3. The airis denser near the earth, than in 
high places ; or denser at the foot of a mountain, than at the 
oe of it. And the higher above the earth, the less dense 
it 18. 3 


357. Corol. 4. The spring ¢ or r elasticity of the air, is diva 
to the weight of the atmosphere above it ; and they will pro- 
duce the same effects : since they always sustain and balance 
each other. 7 
358. Corel. & 


* ke 


ELASTICITY OF AIR. 221 


358. Corol. 5. If the density of the air be increased, pre- 
serving the same heat or temperature its spring or elasticity 
is also increased, and in the same proportion. 


359. Corol. 6. By the pressure and gravity of the atmos- 
phere, on the surface of the fluids, the fluids are made to rise 


in any pipes or vessels, when the spring or pressure within 
is decreased or taken off. 


PROPOSITION LXXITY. 


360. Heat Increases the Elasticity of the Air, and Cold Dimi- 
_mishes it. Or, Heat Expands, and Cold Condenses the Air. 


This property is also proved by experience. 


361. Tuus, tie a bladder very close with some air in. it; 
and Jay it before the fire : then as it warms, it will more and 
more distend the bladder, and at last burst it, if the heat be 
continued and increased high enough. But if the bladder be 
removed from the fire, as it cools it will contract again, as be- 
fore. And it was on this principle that the first air-balloons 


- were made by Montgolfier: for, by heating the air within 


them, by a fire beneath, the hot air distends them to a size 
which occupies a space: in the atmosphere, whose weight of 
common air exceeds that of the balloon. | 


362. Also, if a cup or glass, with a little air in it, be in- 
verted into a vessel of water; and the whole be heated over 
the fire or otherwise ; the air in the top will expand till it fill 
the glass, and expel the water out of it; and part of the air 
itself will follow, by continuing or increasing the heat. 

Many other experiments, to the same effect, might be ad- 
duced, all proving the properties mentioned in the proposition, 


SCHOLIUM. 


363. So that, when the force of the elasticity of air is con- 
sidered, regard must be had to its heat or temperature; the 
same quantity of air being more or less elastic, as its heat is 
more or less. And it has been found, by experiment, that 
the elasticity is increased by the 435th part, for each degree 
of heat, of which there are 180 between the freezing and 
boiling heat, of water. 


364. NV. B. Water expands about the 5,3,,; part, with each 


20000 


degree of heat. (Sir Geo. Shuckburgh, Philos. Trans. 1777, 
p- 5605 &c.) 
Also, 


999 PNEUMATICS. . 


Also, the 
Spec. grav. of air 1:201 or 11.) when the barom, is 29°5, 
water 1000 ‘andthe therm. is 55° 
mercury 10 which are their mean heights 
in this country. 
1 OG 
Or thus, air 1°222 or 4 ieee. baron, is 50, 


water 1000 0 and thermometer 55. 


mercury 13600 
PROPOSITION LXXV. 


365. The Weight or Pressure of the Atmosphere, on any Base at 
the Earth’s Surface, is Equal to ihe Weight of a Column of 
Quicksilver, of the Same Base, and the Height of which is be- 
tween 28 and 31 inches. 


Tuis is proved by the barometer, an instrument which 
measures the pressure of the air, and which is described be- 
low. For, at some seasons, and in.some places, the air sus- 
tains and balances a column of mercury, of about 28 inches : 
but at other times it balances a column of 29, or 30, or near 
31 inches high; seldom in the extremes 28 or 31, but com- 
monly about the means 29 or 30. A variation which depends 
partly on the different degrees of heat in the air near the sur- 
face of the earth, and partly on the commotions and changes 
in the atmosphere, from winds and other causes, by which it 
is accumulated in some places, and depressed in others, being 
thereby rendered denser and heavier, or rarer and lighter ; 
which changes in its state are almost continually happening in 
any one place. But the medium state is commonly about 292 
or 30 inches. 


366. Corol. 1. Hence the pressure of the atmosphere on 
every square inch at the earth’s surface, at a medium, is 
very near 15 pounds avoirdupois, or rather 142 pounds. For, 
a cubic foot of mercury weighing 13600 ounces nearly, an 
inch of it will weigh 7-866 or almost 8 ounces, or nearly half 
a pound, which is the weight of the atmosphere for every 
inch of the barometer on a base of asquare inch; and there- 
fore 30 inches, or the medium height, weighs very near 142 
pounds. , 


367. Corol. 2. Hence also the weight or pressure of the 
atmosphere, 1s equal to that of a column of water from 32 
to 35 feet high, or on a medium 33 or 34 feet high. For, 
water and quicksilver are in weight nearly as 1 to 13-6; 

so 








PRESSURE OF THE ATMOSPHERE. 223 


so that the atmosphere will balance a column of water 13-6 
times as high as one of quicksilver ; consequently | 


13°6 times 28 inches = 381 inches, or 312 feet, 
13:6 times 29 inches = 394 inches, or 328 feet, 
13-6 times 30 inches = 408 inches, or 34 feet, 
13°6 times 31 inches = 422 inches, or 352 feet. 


And hence a common sucking pump will not raise water 
higher than about 33 or 34 feet. And a siphon will not run, 
if the perpendicular height of the top of it be more than about 
33 or 34 feet. 


368. Corol. 3. If the air were of the same uniform den- 
sity at every height up to the top of atmosphere, as at the 
surface of the earth ; its height would be about 51 miles at a 
medium. For, the weights of the same bulk of air "and water, 
are nearly as 1°222 to 1000; therefore as 1:222 : 1000: : 332 
feet : 27600 feet, or 52. miles nearly. And so high the at- 
mosphere would be, if it were all of uniform density, like wa- 
ter. But, instead of that, from its expansive and elastic qua- 
lity, it becomes continually more and more rare, the farther 
above the earth, in a certain proportion, which will be treat- 
ed of below, as ‘also the method of measuring heights ee the 
barometer, which depends on it. 


369. Corol. 4. From this Ss ebhcaition and the last it follows, 
that the height is always the same, of an uniform atmosphere 
above any place, which shall be ‘all of the uniform density 
with the air there, and of equal weight or pressure with the 
real height of the atmosphere above that place, whether it be 
at the same place, at different times, or at any different places 
or heights above the earth ; and that height is always about 
51 miles, or 27600 feet, as above found. For, as the density 
varies in exact proportion to the weight of the column, there- 
fore it requires a column of the same height in all cases, 
to make the respective weights or pressures. Thus, if w 
and w be the weights of atmosphere above any places, p and 
d their densities, and 4 and h the: heights of the uniform 
columns, of the same densities and weights; Then u x p 
=w,andh X d=w; therefore = or His equal to or h. The 
temperature being the same. 


PROPOSITION 


224 : -. PNEUMATICS. 
PROPOSITION LXXVI. _ 


370. The Density of the Atmosphere, at Different Heights above 
the Earth, Decreases in such Sort, that when the Heights In- 
crease in Arithmetical Progression, the Densities Decrease in . 
Geometrical Progression. ) | 


. Ler the indefinite perpendicular line ar, | 
erected on the earth, be conceived to be divid- 
ed into a great number of very small equal 
parts, aA, B, c, p, &c. forming so many thin stra- 
ta of air inthe atmosphere, all of different den-— 
sity,’ gradually decreasing from the greatest at 
A: then the density of the several strata, a, B, 
c, D, &c. will be in geometrical progression de- 
creasing. 

For, as the strata a, B, c, &c. are all of equal 
thickness, the quantity of matter in each of them, is as the 
density there; but the density in any one, being as the com- 
pressing force, is as the weight or quantity of all the matter 
from that place upward to the top of the atmosphere; there- 
fore the quantity of matter in each stratum, is also as the 
whole quantity from that place upward. -Now if from the 
whole weight at any place as B, the weight or quantity in the 
stratum B be subtracted, the remainder is the weight at the 
next stratum c; that is, from each weight subtracting a part 
which is proportional to itself, leaves the next weight; or, 
which is the same thing, from each density subtracting a part 
which is proportional to itself, leaves the next density. But 
when any quantities are continually diminished by parts which 
are proportional to themselves, the remainders form a series 
of continued proportionals : consequently these densities are 
in geometrical progression. eS et 

Thus, if the first density be p, and from each be taken 
n—1 








its nth part ; there will then remain its part, or the — 
; n 


part, putting m for n—1,; and therefore the series of den- 
ee m m* m?_ m4 | Nh 
sities will be p, —»p,— p,— v,—- p, &c. the comman ratio 
ihe nm  n2 n3 nm’ 
of the series being that of n to m. 

SCHOLIUM. 


371. Because the terms of an arithmetical series, are. pro- 
portional to the logarithms of the terms of a geometrical 
series: therefore different altitudes above the earth’s sur- 

: face, 


DENSITY OF THE ATMOSPHERE, 225 


face, are as the logarithms of the densities, or of the weights 
of air, at those altitudes. 
So that, if p denote the density at the altitude a, 
andd - the density at the altitude a; _ 
then a being as the log. of pv, and a as the log. of d, 


the dif. of alt. a—a will be as the Jog. p—log. d. or log-~. 


And if a=0, or D the density at the surface of the earth ; 
D 
d 


Or, in general, the log. of - is as the altitude of the one place 


then any altitude above the surface a, is as the log. of 


above the other, whether the lower place be at the surface of 
the earth, or any where else, : 


And from this property is deriyed the method of determine, 
ing the heights of mountains and other eminences, by the ba- 
rometer, which is an instrument that measures the pressure or 
density of the air at any place. For, by taking with this inr 
strument, the pressure or density, at the foot of a hill for in- 
stance, and again at the top of it, the difference of the loga- 
rithms of these two pressures, or the logarithm of their quo- 
tient, will be as the difference of altitude, or as the height of 
the hill; supposing the temperatures of the air to be the 
Same at both places, and the gravity of air not altered by the 
different distances from the earth’s centre, : 


372. But as this formula expresses only the relations be- 
tween different altitudes with respect to their densities, re- 
course must be had to some experiment, to obtain the real al- 
titude which corresponds to any given density, or the density 
which corresponds to a given altitude. And there are various 
experiments by which this may be done. The first, and most 
natural, is that which results from the known specific gravity 
of air, with respect to the whole pressure of the atmosphere 
on the surface of the earth. Now, as the altitude a is always 


as log. - assume h so that ash Xlog. 5, where h will be of 


one constant value for all altitudes ; and to determine that va- 
lue, let a case be taken in which we know the altitude a cor- 
responding to a known density d; as for instance, take a=1 
foot, or I inch, or some such smail altitude ; then, because the 
density p may be measured by the pressure of the atmosphere, 
_or the uniform column of 27600 feet, when the temperature 
is 55°; therefore 27600 feet will denote the density p at the 
You, Il. 30 - lower 


296 “PNEUMATICS. 


lower place, and 27599 the less density d at 1 foot above it? 


consequently 1=hXlog. og a3 which, by the nature of lo- 


43429448 h 
garithms, is nearly = =hx “o7600 7 = BSEEI - 


hence h = 63551 feet ; which gives, for any altitude in gene- 


sauna ; and 


ral, this theorem, viz. a= 63551 X log. “2 or = 63551 X log. 


= feet, or 10592 Xlog. = fathoms ; where m is the column of 


‘mercury which is equal to the pressure or weight of the at- 
mosphere at the bottom, and m that at the top of the altitude 
a; and where mand m may be taken in any measure, either feet 
or inches, &c. 


‘373. Note, that this formula i is adapted to the mean tem- 
perature of the air 55°. But, for every degree of tempera- 
ture different from this, in the medium between the tempera- 
tures at the top and bottom of the altitude a, that altitude will 
vary by its 435th part; which must be added, when that me- 
dium exceeds (55°, otherwise subtracted. 


374. Note, also, that a column of 30 inches of mercury 
varies its length by about the 31; part of an inch for every 
degree of heat, or rather ;,/,, of the whole volume. 


375. But the formala may be rendered much more con- 
venient for use, by reducing the factor 10592 to 10000, by © 
changing the temperature proportionally from 45°; thus, 
as the diff. 592 is the 18th part of the whole factor 10592 ; 
‘and as 18 is the 24th part of 435; therefore the correspond- 
ing change of temperature is' 24°, which reduces the 55° to 


_ 31%. 8o that the formula is, a = 10000 x log. — . fathoms, 


when the temperature is 31 degrees; and for sary degree 
above that, the result is to be increased by so many times its 
435th part. 


376. Exam. 1. To find the height of a hill when the pres- 
sure of the atmosphere is equal to 29:68 inches of mercury 
at the bottom, and 25:28 at the top; the mean temperature 
being 50° ? Ans. 4878 feet, or 730 fathoms. 


377. Exam. 2. To find the height of a hill when the at- 
mosphere weighs 29-45 inches of mercury at the bottom, and 
26°82 at the top, me mean temperature being 33° ? : 

Ans. 2385 feet, or 3974 fathoms. 
378. Exam, 3. 


THE SIPHON. 227 


378. Exam. 3. At what altitude is the density of the at- 
mosphere only the 4th part of what it is at the earth’s sur- 
face? Ans. 6020 fathoms. 

By the weight and pressure of the atmosphere, the effect 
and operations of pneumatic engines may be accounted for, 
and explained ; such as siphons, pumps, barometers, &c. ; of 
which it may not be improper here to give a brief Resemplion. 


OF THE SIPHON. 


379. Tue Siphon, or Syphon, is any 
bent tube, having its two legs either of 
equal or of unequal length. ye 

If it be filled with water, and ent in- 
verted, with the two open ends down- 
ward, and held level in that position ; 
the water will remain suspended in it, 
if the two legs be equal. For the at- 
mosphere will press equally on the sur- 
face of the water-in each end, and sup- 
port them, if they are not more than 34 feet high: and the 
Jegs being equal, the water in them is an exact counterpoise 
by their equal weights ; so that. the one has no power to 
move more than the other ; ; and tiey are both supported bY 
the atmosphere. 

But if now the siphon be a little caokitied tb one side, so 
that the orifice of one end be lower than that of the other ; 
or if the legs be of -unequal length, which is the same thing ; 
then the equilibrium is destroyed, and the water will all de- 
scend out by the lower end, and rise up in the higher. For, 
the air pressing equally, but the two ends weighing unequally, 
a motion must commence where the power is greatest, and so 
continue till all the water has run out by the lower end. . And 
if the shorter leg be immersed into a vessel of water, and the 
‘siphon be set a running as above, it will continue to run till 
all the water be exhausted out of the vessel, or at least as low 
as that end of the siphon. Or, it may be set a running with- 
out filling the siphon as above, by only inverting it, with its 
shorter leg into the vessel of water; then, with the mouth 
applied to the lower orifice a, suck the air out ; and the wa- 
ter will presently follow, being forced up in the siphon by the 
pressure of the air on the water in the vessel. ob. 








f 228 ] 
OF THE PUMP. 


380. Pient ate three sorts G8 , a 
of pumips: the Sucking, the — ty . 
cb m> me 






Lifting, and the Forcing Pump. AE 
By the first, water canb@ raised 
only to about 34 feet. viz. by 
the pressure of the atmos- 
phere; but by the others; to 
any height ; but then they re- 
quire more apparatus and pow- 
er: i 
The aniiexed figure repre- | 
sents a common sucking pump: 
AB is the barrel of the pump, 
being a hollow cylinder, made 
of metal, and smooth within, or 
of wood ‘for véry common pur- 
poses, Cp is the handle, move- 
able about the pin &, by moving 
the énd c up and down. vr 
ad iron tod turning about a pin, === 
b; which connects it to the end of the handle. This rod is 
fixed to the piston, bucket, or sucker, Fre, by which this is 
iioved up and down within the barrel, which it must fit very. 
tight and close, that no air or water may pass between the 
piston and the sides of the barrel; and for this purpose it is 
comimonly armed with leather. ‘The piston 1s made hollow, 
or it has a perforation through it, the orifice of which is co- 
vered by a valve n opening upwards. 1is a plug firmly fixed 
in thé lower part of the barrel, also perforated, and covered 
by a valve « opening. upwards. 
381. When the pump is first to be worked, and the water 
is below the plug 1; raise the end c of the handle, then the . 
piston descending, compresses the air in HI, which by its 
spring shuts fast the valve x; and pushes up the valve u, 
and so enters into the barrel above the piston. Then put: 
ting the end c of the handle down again, raises the piston 
or sucker, which lifts up with it the column of air above it, 
the external atmosphere by its pressure keeping the valve H 
shut: the air in the barrel being thus exhausted, or rarefied, 
is no longer a couriterpoise to that which presses on the sur- 
face of the water in the well ; this is forced up the pipe, and 
through the valvé «, into the barrel of the pump. Then 
pushing the piston down again into this water, now - 2% 
arre 


THE AIR-PUMP. 229 


barrel its weight shuts the lower valve xk, and its resistance 
forces up the valve of the piston, and enters the upper part 
of the barrel, above the piston. Then, the bucket being 
raised, lifts up with it the water which had passed above its 
valve, and it runs out by the cock 1} and taking off the 
weight below it, the pressure of the external atmosphere on 
the water in the well again forces it up through the pipe and 
lower valve close to the piston, all the way as it ascends, thus 
keeping the barrel always fullof water. And thus by repeat- 
ing the strokes of the piston, a continued discharge is made 
dt the cock L; : 


OF THE AIR-PUMP. 


382, NeArty on the same principles as the water pump, 
ig the invention of the Air-pump, by which the air is drawn 
out of any vessel, like as water is drawn out by the former. 
A brass barrel is bored and polished truly cylindrical, and ex- 
actly fitted with a turned piston, so that no air can pass by 
the sides of it, and furnished with a proper valve opening 
upward. Then by lifting up the piston, the air in the close 
vessel Lelow it follows the piston, and fills the barrel; and 
being thus diffused through a larger space than before, when 
it occupied the vessel or receiver only, but not the barrel, 
it 1s made rarer than it was before, in proportion as the ca- 
pacity of the barre] and receiver together exceeds the re- 
ceiver alone. Another stroke of the piston exhausts another 
barrel of this now rarer air, which, again rarefies it in the 
same proportion as before. And so on, for any number of 
strokes of the piston, still exhausting in the same geometri- 
cal progression, of which the ratio is that which the capacity 
of the receiver and bariel together exceeds the receiver, till 
this is exhausted to any proposed degree, or as far as the na- 
ture of the machine is capable of performing ; which happens 
when the elasticity of the included air is so far diminished, by 
rarefying, that it is too feeble to push up the valve of the pis- 
ton and escape. i | 

383. From the nature of this exhausting, in geometrical 
progression, we may easily find how much the air in the re- 
ceiver is rarefied by any number of Strokes of the piston ; or 
what number of such strokes is necessary, to exhaust the re- 
ceiver to any given degree. Thus, if, the capacity of the re- 
-seiver and barrel together, be to that of the receiver alone, 
as 


230 OF THE DIVING BELL. 


asc to r, and 1 denote the natural atin of the air at: first: 
then 


ye prevl i, the denaity bike one stroke of the piston, » 
Pea lis Ld Us the denaly atte! 
pie IS Go the density after 2 strokes, 

Ra 


sate athe, the density after three strokes, 


—&e. and — the density after n strokes. 
So, if the barrel be equal to } of the receiver ; thence: 7: 
An i f . 
5:4; and ep OP is = d the density after » turns. And 


if n be 20, then 0°82°= -0115 is the density of the included 
air after 20 strokes of the piston ; which being the 86,7, part 
of 1, or the first density, it follows that the air is 8655 times 
rarefied by the 20 strokes. 

384. Or, if it were required to find the number of strokes 
| he to rarefy the air any number of times ; because 


= is = the aid ha density d; therefore, taking the loga- 


log. d 
| lr—lic’ 
strokes required. So if r be 4 of c, and it be required to 
rarify the air 100 times: then d= j;1, or 0-1: and hence 


log. 100 . , : 
ae = aeyry ns 203 nearly. .So that in 202 strokes the air 
will be rarefied 100 times. 


rithms, ax bop Pa log. d, and n = the number of 





OF THE DIVING BELL & CONDENSING MACHINE. 


- 885. On the same principles too depend the operations 
and effect of the Condensing Engine, by which air may be 
condensed to any degree instead of rarefied as in the air-pump. 
And, like as the air-pump rarefies the air, by extracting always 
one barrel of air after another ; so by this other machine, the 
air 1s condensed, by throwing in or adding always one barrel 
of air after another ; which it is evident may be done by only 
-furning the valves of the piston and barrel, that is, making 
them to open the contrary way, and working the piston in the 
fame manner ; so that ‘as they both open upward or outward in 

the 


OF THE DIVING BELL. 231 


the air- pump or rarefier, they. will both open. downward or 
inward in the condenser. 

386. And on the same principles, namely, of the compres- 
sion and. elasticity of the air, depends the use of the Diving 
Bell, which is a large vessel, in which a person descends to 
the bottom of the sea, the open end of the vessel being down- 
ward ; only t in this case the air is not condensed by forcing 
more of it into the same space, as in the condensing engine ; 
but by compressing the same quantity of air into a less space 
in the bell, by increasing always the force which compresses it. 

387. If a vessel of any sort be inverted into water, and 
pushed or let down to any depth in it; then by the pressure 
of the water some of it will ascend into the vessel, but not so 
high as the water without, and will compress. the air into less 
space, according to the difference between the heights of the 
internal and external water; and the density and elastic force 
of the air will be increasedin the same proportion, as its space 
in the vessel is diminished. 

So, if the tube cr be inverted, and pushed down into wa- 
ter, till the external water exceed the internal, by the height 
as, and the air of the tube be réduced to the space cp ; ther 
that air is pressed both by a column of 
water of the height az, and by the whole . 
atmosphere, which presses on the upper 
surface of the water; consequently the — 
space cp is to the whole space. cz, as the 
weight of the atmosphere, is to the weights 
both of the atmosphere and the column 
of water as. Sothatif ap be about 34 . 
feet, which is equal to the force of the 
atmosphere, then cp will be equal to 4cz ; 
but if as be double of that, or 68 feet, 
then cp will be ice; andsoon. And hence, by knowing the 
depth ar, to which the vessel is sunk, we can easily find the 
point p, to which the water will rise within it at any time. For 
let the weight of the atmosphere at that time be equal to that 
of 34 feet of water ; also, let the depth ar be 20 feet, and 
the length of the tube cz 4 re then putting the height of 
the internal water abn, 

. itis S4-+as: 2 CE:3 6b, 

that i is. bein 34: ‘CE ! CE—DE, 

or 54—27: 34::4:4—2; 
hence, multiplying extremes and means, 216—-58x-+-22 = 136, 
and the root is z= ,/2 very nearly =1-414 of a foot, or 17 
inches nearly ; being the height pe to which the water will 
rise within the tube. 





388. But 


232 | THE BAROMETER, 


388. But if the vessel be not 
equally wide througheut, but of 
any other shape, as of abell-like , 
form, suchasis usedin diving; then 
the altitudes will not observe the — 
proportion above, but the spaces. - 
or bulks only will respect that Po 
-portien, namely, 34-+-aB : 34; : ca- 
pacity cKL.; capacity cHI, if it be 
common or: fr ésh water ; and 3$+- 

AB: 33: : capacity cKL : capacity 
cui, if it be sea-water. From 
which proportion, the height pz may be found, when ther ng: 
ture or shape of the vessel or bel] cxx is kitown: 





OF THE BAROMETER. 


389. THE Barometer is an instrument for measuring the 
pressure of the atmosphere, and elasticity of the air, at any 
time. It is commonly made of a glass tube, of near 3 feet 
long, close at one end, and filled with mercury. When the 
tube is full, by stopping the open end with the finger, then in- 
_ verting the tube, and immersing that end with the finger inte 
a bason of quicksilver, on removing the finger from the ori- 
fice, the fluid in the tube will descend into the bason, till what 
remains in the tube be of the same weight with a column of 
the atmosphere, which is commonly between 28 and 31 inches 
of quicksilver ; and leaving an entire vacuum in the upper 
end of the tube above the mercury. For, as the upper end 
of the tube is quite void of air, there is no pressure down- 
wards but from the column of quicksilver, and therefore that 
will bean exact balance to the counter pressure of the whole 
column of atmosphere, acting on the orifice of the tube by the 
quicksilver in the bason. The upper 3 inches of the tube, 
namely, from 28 to 31 inches, have a scale attached to them, 
divided into inches, tenths, and hundredths, for measuring the 
length of the column at all times, by observing which division 
of the scale the top of the quicksilver is opposite to ; as it as- 
cends and descends within these limits according to the state 
of the atmosphere. 

Se 


THE THERMOMETER. 033 


So that the weight of the quick- 
silver in the tube, above that in 


the bason, is at all times equal to 

the weight or pressure of the co- 

lumn of atmosphere above it, and 

of the same base with the tube ; bade 
and hence the weight of it may | “I 


at all times be computed; being 
nearly at the rate of halfa pound 
avoirdupois for every inch of quick- 
silver in the tube, on every square 
inch of base; or more exactly it 
is 5°, of a pound on the square 
inch, for every inch in the altitude 
of the quicksilver weighs just ,£2,Ib, 
or nearly } a pound, in the mean 
temperature of 55° of heat. And , 
consequently, when the barometer © 
stands at 30 inches, or 21 feet high, | 
which is nearly the medium or 
standard height, the whole pressure 197 
of the atmosphere is equal to 143 

pounds, on every square inch of the base ; andso in propor- 
tion for other heights. | 


1 


= on 
: Se 


Soar Oy -AROe Pe 








&S 
& 





RRR =; 


iy area 
OF THE THERMOMETER. 


390. THE Tuermomerer is an instrument for measuring 
the temperature of the air, as to heat and cold. 

It is found by experience, that all bodies expand by heat, 
and contract’ by cold; and hence the degrees of expansion. 
become the measure of the degrees of heat. Fluids are 
more convenient for this purpose than solids; and quick- 
silver is now most commonly used for it. A very fine glass 
tube, having a pretty large hollow ball at the bottom, is 
filled about half way up with quicksilver: the whole being 
then heated very hot till the quicksilver rise quite to 
the top, the top’ is then hermetically sealed, so as perfectly 
to exclude all communication with the outward air. Then, 
in cooling, the quicksilver contracts, and consequently its 
surface descends in the tube, till it come to a certain point, 
_correspondent to the temperature or heat of the air. And 
when the weather becomes warmer, the quicksilver expands, 

Vor. Il. 31 and 


oS4 “PHE THERMOMETER. 


and its surface rises in the tube ; and 
again contracts and descends when the 
weather becomes cooler. So that, by 
placing a scale of any divisions against 
the side of the tube, it will show the 
degrees of heat by the expansion and — 
contraction of the quicksilver in the 
tube ; observing at what division of the 
scale the top ef the quicksilver stands, 
And the method of preparing the scale, 
as used in England, is thus :—Bring the 
thermometer into the temperature of 
freezing, by immersing the ball in water 
just freezing, or in ice just thawing, and 
mark the scale where the mercury then: 
stands, for the point of freezing. Next, 
immerge it in boiling water; and the 
quicksilver will rise to a certain height 
in the tube; which mark also on the 
scale for the boiling point, or the heat 
of boiling water. Then the distance be- 
tween these two points, is divided into 
180 equal divisions, or degrees ; and the 2 

like equal degrees are also continued to any extent below the 
freezing point, and above the boiling point. The divisions 
are then numbered as follows ; namely at the freezing point 
is set the number 32, and consequently 212 at the boiling 
point ; and all the other numbers in their order. 





This division of the scale is commonly called Fahrenheit’s. 
According to this division, 55 is at the mean temperature of 
the air in this country ; and it is in this temperature, and in 
an atmosphere which sustains a column of 30 inches of quick- 
silver in the barometer, that all measures and specific gravities 
are taken, unless when otherwise mentioned ; and in this tem- 
perature and pressure the relative weights, or specific gravi- 
ties of air, water and quicksilver, are as 

12 for air and these also are the weights of a cu- 
1000 for water, i foot of each, in avoirdupois ounces, 
136C0 for mercury ; (in that state of the barometer and ther- 
mometer. For other states of the thermometer, each of these 
bodies expands or centracts according to the following rate, 
with each degree of heat, viz. 


Air about - 735 part of its bulk, 
Water about woes ; part of its bulk, 
Mercury about 5,55 part of its bulk. 


ON 


[i256 


ON THE MEASUREMENT OF ALTITUDES BY THE 
BAROMETER AND THERMOMETER. 


391. FROM the principles laid dewn in the scholium to 
prop. 76, concerning the measuring of altitudes by the baro- 
meter, and the foregoing descriptions of the barometer and 
thermometer, we may now collect together the precepts for 
the practice of such measurements, which are as follow : 


First. Observe the height of the barometer at the bottom 
of any height, or depth, intended to be measured; with the 
temperature of the quicksilver, by means of a thermometer 
attached to the barometer, and also the temperature of the 
air in the shade by a detached thermometer. 


Secondly. Let the same thing be done also at the top of the 
said height or depth, and at the same time, or as near the 
same time as may be. And let those altitudes of barometer 
be reduced to the same temperature, if it be thought neces- 
sary, by correcting either the one or the other, that is, aug- 
ment the height of the mercury in the colder temperature, 
or diminish that in the warmer, by its 5,1,, part for every de- 
gree of difierence of the two. 

Thirdly, ‘Take the difference of the common logarithms of 
the two heights of the barometer, corrected as above if ne- 
cessary, cutting off 3 figures next the right hand for decimals, 
when the log-tables go to 7 figures, or cut off only 2 figures 
when the tables go to 6 places, and so on; or in general re- 
move the decimal point 4 places more towards the right hand, 
those on the left hand being fathoms in whole numbers. 


Fourthly. Correct the number last found for the. difference 
of temperature of the air, as follows; Take half the sum of 
the two temperatures, for the mean one: and. for every de- 
gree which this differs from the temperature 31°, take so 
many times the ;1, part of the fathoms above found, and add 
them if the mean temperature be above 31°, but subtract them 
if the mean temperature be below 31°; and the sum or dif- 
ference will be the true altitude in fathoms: or, being multi- 
plied by 6, it will be the altitude in feet. . 


392. Example 1. Let the state of the barometers and 
thermometers be as follows ; to find the altitude, viz. 





Baroma. Thermom, | 
attach. | detach. Ans. the alt. is 
Lower29-68 57 57 7191 fathoms. 
Upper25-28 | 43 42 | | 








393. Exam, 


236 THE RESISTANCE OF FLUIDS, &c. 


393. Exam. 2. To find the altitude, when the state of the 
barometers and thermometers is as follows, viz. 





‘Barom. Thermom. aa 
attach. | detach. Ans. the alt. is 
Lower29°45 38 31 409 ,8, fathoms, 
Upper26°82 41 35 or 2458 feet. 
eo 


ON THE RESISTANCE OF FLUIDS, WITH THEIR 
FORCES AND ACTIONS ON BODIES. 


PROPOSITION LXXVII. 


394. If any Body Move through a Fluid at Rest, or the Fluid 
Move against the Body at Rest; the Force or Resistance of 
the Fluid against the Body, will be as the Square of the Ve- 
locity and the Density of the Fluid. That ws, r« dv? . 


For, the force ow resistance is as the quantity of matter or 
particles struck, and the velocity with which they are struck. 
But the quantity or number of particles struck in any time, 
are as the velocity and the density of the fluid. Therefore 
the resistance, or force of the fluid, is as the density and 
square of the velocity. 

395. Corol. 1. The resistance to any plane, is also more 
or less, as the plane is greater or less : and therefore the ree 
sistance on any plane, is as the area of the plane a, the den- 
sity of the medium, and the square of the velocity. That is, 
RXadv?. 

396. Corol. 2. If the motien be not perpendicular, but 
oblique to the plane, or to the face of the body; then the 
resistance, in the direction of motion, will be diminished in 
the triplicate ratio of radius to the sine of the angle of incli- 
nation of the plane to the direction of the motion, or as the 
cube of radius to the cube of the sine of that angle. So that 
Ra adv*s?, putting 1 = radius, and s = sine of the angle of 
inclination cas. 

_. Fer,if as be the plane, ac the 
direction of motion, and gc perpen- 
dicular to ac ; then no more particles 
meet the plane than what meet the 
perpendicular 8c, and therefore their ' 
number 1s diminished as az to Bc or 
as 1tos. But the force of each par 





ticle, 


THE RESISTANCE OF FLUIDS, &c. 237 


ticle, striking the plane obliquely in the direction ca, is also 
diminished as a8 to Bc, or as 1 tos; therefore the resistance, 
which is perpendicular to the face of the plane by art 52, is 
as 12 tos?. But again, this resistance in the direction perpen- 
dicular to the face of the plane, is to that in the direction ac, 
by art 51, as an tose, orasitos. Consequently, on all these 
accounts, the resistance to the plane when moving perpendi- 
cular to its face, is to that when moving obliquely, as 1% to s°, 
or 1tos?. That is, the resistance in the direction of the 
motion, is diminished as 1 to s%, or in the triplicate ratio of 
radius to the sine of inclination. 


PROPOSITION LXXVIII. 


397. The Real Resistance to a Plane, by a Fluid acting in a 
Direction perpendicular to its Face, 1s equal to the Weight 
of a Column of the Fluid, whose Base is the Plane, and Al- 
titude equal to that which ts due to the Velocity of the Motion, 
or through which a Heavy Body must fall to acquire that Ve- 
locity. | | 


Tue resistance to the plane moving through a fluid, is the 
game as the force of the fluid in motion with the same veloci- 
ty, on the plane at rest. But the force of the fluid in motion, 
is equal to the weight or pressure which generates that motion ; 
and this is equal to the weight or pressure of a column of the 
fluid, whose base is the area of the plane, and its altitude that 
which is due to the velocity. En 

398. Corol. 1. If a denote the area of the plane, v the 
velocity, the density or specific gravity of the fluid, and 
g§ = 16, feet, or 193 inches. Then the altitude due to the 

2 2 m2 
velocity v being Kis therefore a Xn X =~ will be the 
} 4g’ 4g 4g 
whole resistance, or motive force R. 

399. Corol. 2. lf the direction of motien be not perpen- 
dicular to the face of the plane, but oblique to it, in any an- 
gle, whose sine is s. Then the resistance to the plane will be 
anv 33 

4g 
400. Corol. 3. Also, if w denote the weight of the body, 
whose plane face a is resisted by the absolute force r ; then 
any? 33 
. Asw 
401. Corol. 4. And if the body be a cylinder, whese face 

Pag or 





— 





the retarding force f, or = will be, 


238 THE RESISTANCE OF FLUIDS, &c. 
or end is a, and radius r, moving in the direction of its axis ; 
lig then s = 1, and a = pr?, where Es 1416; then 


he rer —vall be the resisting force r, yf ie’ ane the retarding 
Sw 








force f. 
402. Corol. 5. This is the value of the resistance when 


the end of the cylinder is a plane perpendicular to its axis, 
or to the direction of motion. But were its face an elliptic 
section, or a conical surface. or any other figure every where 
equally inclined to the axis, or direction of motion, the sine 
or inclination being s: then, the number of particles of the 
fluid striking the face being still the same, but the force of 

each opposed to the direction of motion, diminished in the 
duplicate ratio of radius to the sine of inclination, the resist- 
pnr? v2 33 


oe 
® 


5 





ing force zk would be 


PROPOSITION LXXIX. 


403. The Resistance to a Sphere moving through a Fluid, is 
but Half the Resistance to tts Great Circle, or to the End of a 
Cylinder of the same pe gS. moving with an Equal Velo- 


city. 


Ler ares be half the sphere, moving 
in the direction crc. Describe the para- 
boloid arzgxs on the same base. Letany | 
particle of the medium meet the semicir- 
cle in F, te which draw the tangent re, 
the radius Fc, and the ordinate rrm. Then 
the force of any particle on the surface at 
F, is to its force on the base at m, as the 
square of the sine of the angle gc, or its 
equal the angle rcu, to the square of radius, that is, as 
uF? to cr?. ‘Therefore the force of all the particles, or the 
whole fluid, on the whole surface, is to its force on the circle 
of the base, as al] the nr? to as many times cr?.. But cr? 
is = ca? = ac. cp, and HF? = an. ue by the nature of. the 
circle : also, Au. HB: AG .-CB: : HI: cE by the nature of the 
parabola: consequently the force on the epherical surface, is 
to the force on its circular base, as all the n1’s to as many cr’s, 
that is, as the content of the paraboloid to the content of its 
circumscribed cylinder, namely as | to 2. 

yeh Corol. Hence, the resistance to the sphere isn = 


iba , being the half of that of a cylinder of the same 
ig diameter, 








siipcae GRAVITY. 239 


diameter. For example, a -Olb i iron ball, whose diameter is 4 
- inches, when moving through the air with a velocity of 1600 
feet per second, would meet a resistance which is equal to a 
weight of 1322lb, over and above the pressure of the atmos- 
phreae for want of the counterpoise behind the wall. 


-PRACTICAL EXERCISES CONCERNING SPECIFIC 
GRAVITY. 


The Specific Gravities of Bodies are their relative weights 
contained under the same given magnitude ; as a cubic foot, 
or a cubic inch, &c. 

The specific gravities of several sorts of matter, are @X- 
pressed by the numbers annexed to their names in the Table 
of Specific Gravities, at page 211; from which the numbers 
are to be taken, when wanted. 

Note. The several sorts of wood are supposed to be dry. 
Also, ag a cubic foot of water weighs just 1000 ounces avoir- 
dupois, the numbers in the table express, not only the specific 
gravities of the several bodies, but also the weight of a cubic 
foot of each in avoirdupois ounces ; and hence, by proportion, 
the weight of any other quantity, or the quantity of any other 
weight, may be known, as in the dellowing problems. | $ 


PROBLEM I, 
To find the Magnitude of any Body, from its Weight 


As the tabular specific gravity of the body, 

Is to its weight in avoirdupois ounces, 

So is one cubic foot, or 1728 cubic inches, 

To its content in feet, or inches, respectively. 


EXAMPLES. 


‘Exam, 1. Required the content of an irregular block of 
common stone, which weighs 1 cwt or 112lb. 
: Ans. 12284 cubic inches. 
Bxax. 2. How many cubic inches of gunpowder are there 


in Ilb weight ! 2 Ans. 291 cubic inches nearly. 
Exam. 3: How many cubic feet are ‘there in a ton weight 
of dry oak ? Ans. 38122 cubic feet. 


PROBLEM 


240 SPECIFIC GRAVITY. 


PROBLEM Il. 


To Find the Werght of a Body from wuts Magnitude. 


As one cubic foot, or 1728 cubic inches, 
Is to the content of the body, 

So is its tabular specific gravity, 

To the weight of the body. 


EXAMPLES. 


isin. 1. Required the weight of a block of marble, whose 
length is 63 feet, and breadth and thickness each 12 feet ; 
being the dimensions of one of the stones in the walls of 
Balbeck ? 
Ans. 6834, ton, which is nearly equal to the burden 
of an East- {India ship. 
Exam. 2. What is the weight of 1 pint, ale measure, of 
gunpowder ? Ans. 19 oz nearly. 
Exam. 3. What is the weight of a block of dry oak, which 


measures 10 feet in length, 3 feet broad, and 21 feet deep ; : 
Ans. 43351 2b. 


PROBLEM ITI. 


To Jind the Specific Gravity of a Body. 


Casr 1. When the body is heavier than water, weigh it 
both in water and out of water, and take the difference, which 
will be the weight lost in water. Then say, 

As the weight lost in water, 

Is to the whole weight, 

So is the specific gravity of water, 
To the specific gravity of the body. 


EXAMPLE. 


A piece of stone weighed 10lb, but in water only 621b, re- 
quired its specific gravity ? Ans. 2609. 

Case 2. When the body is lighter than water, so that it will 
not quite sink, affix to it a piece of another body, heavier 
than water, so that the mass compounded of the two may sink 
together. Weigh the denser body and the compound mass 
separately, both in water and out of it; then find how much 
each loses in water, by subtracting its ‘weight i in water from 
its weight in air; and subtract the less of these remainders 


from the greater. Then say, 
As 


| SPECIFIC GRAVITY. 24h 


As the last Besta dice: ee 

Is to the weight of the light body In air, 
So is the specific gravity of water, 

To the specific gravity of the ree: 


EXAMPLE. 


Suppose ; a piece of elm weighs 151b inair ;: and thata piece 
of copper which weighs 18lb in air. and 16!b in water, is affix- 
ed to it, and that the compound weighs 61b in water ; required 
the specific gravity of the elm? Ans. 600, 


PROBLEM IV. 

To find the Quantities of Two Ingredients, in a given Compound, 

Taxe the three differences of every pair of the three 
specific gravities, namely, the specific gravities of the com- 
pound and each ingredient; and multiply the difference of 
every two specific gravities by the third. Then say, as the 
greatest product, is to the whole weight of the compound, 
so is each of the other products, to the two weights of the in- 


| a 
EXAMPLE. 

A composition of 112lb being made of tin and copper, 
whose specific gravity if found to be 8784; required the 
quantity of each ingredient, the specific Sry 9 of tin being 
7320, and of copper 9000 ? 

Ans. there is 100lb of copper 2. ve 
and consequently 12lb_ of tin ¢ in the composition. 


cepa 


OF THE WEIGHT AND DIMENSIONS OF BALLS AND 
- SHELLS. 


Tse weight and dimensions of Balls and Shells sah be 
found from the probiems last given, concerning specific gravi- 
ty. But they may be found still easier by means of the ex- 
perimented weight of a ball of a given size, from the known 
proportion of similar figures, aiid as the cubes of their 
diameters, 


- PROBLEM J, 


Po find ‘he: Weight of an Iron Ball fr om if Diameter. 


An ‘iron hall of 4 inches diameter | weighs Sib, and the 
weights being as the cubes of the diamieters, it vill be, as 64 
Vor. i. | 39 : : Siald: | 


* 


249 BALLS AND SHELLS. 


(which is the cube of 4) is to 9 its weight, so is the cube of 
the diameter, of any other ball, toits weight. Or, take 2% of 
the cube of the diameter, for the weight. Or, take } of the 
cube of the diameter, and } of that again, and add the two 
together, for the weight. 


EXAMPLES. 
_ Exam. 1. The diameter of an iron shot being 6-7 inches, 
required its weight ? Ans. 42-294]b. 


Exam. 2. What is the weight of an iron ball, whose diame- - 
“ter is 5°54 inches? Ans. 24lb nearly. 


-. PROBLEM i 
To find the Weight of a Leaden Ball. 


~ Aleaden ball of one inch diameter weighs #4 73; of ald; there-. 


fore as the cube of 1 is to 4, or as 14 is to 3, so is the cube 


of the diameter of a teaden ball, to its weight. Or, take 2 - of 
the cube of the diameter, for the weight, nearly. 


EXAMPLES. 
Exam. 1. Required the weight of a leaden ball of 6-6 inches 
diameter ? Ans. 61-°606lb. 
Exaw. 2. What is the weight of a leaden ball of 5°30 in- 
ches diameter? — Ans. 32lb nearly. 
PROBLEM Iil. 
To find the Diameter of an Iron Ball. 
Mortirvy the weight by 71, and the cube root of the pro- 
duct will be the diameter. 
EXAMPLES. 


Exam. 1. Required the diameter of a 491 i iron ball ? 
Ans. 6-685 inches. 

Exam. 2. What is the diameter of a 24lb iron ball? - 
Ans, 5:54 inches. 


PROBLEM IV. — 
To find the Diameter of a Leaden Ball. 


‘Muntipry the weight by 14, and divide the product by 3; 
‘then the cube root of ‘the quotient will be the diameter. | 


EXAMPLES. 


' BALLS AND SHELLS. 245 
‘EXAMPLES. . 


Exam. 1, "Required the diameter of a 641b leaden ball ? 
Ans. 6°684 inches, 

Exam. 2. What i is the diameter of an 8lb leaden ball ? 
Ans. 3° 348 inches. 


PROBLEM V. 
To find the Weight of an Iron Shell. 


‘Tax ,; of the difference of the cubes of the external and 
internal diameter, for the weight of the shell. | 

That is, from the cube of the external diameter, take the 
eube of the internal diameter, winters the remainder by 9, 
and divide the product by 64. 


EXAMPLES. 


Exam. 1. The outside diameter of an iron shell being 12°8, 
and the inside diameter 9-1 inches ; required it weight? 
Ans. 188-94 1Ib. 
Exam. 2. What is the weight of an iron shell, whose ex- 
ternal and internal diameters are 9°8 and 7 inches 7 a 
; ae ‘Ans. 84i1b. 
PROBLEM VI. 


To find how much Powder will fill a Shell. 


Divine the cube of the internal diameter, in inches: by 
57°3, for the pounds of powder. 


EXAMPLES, 


Exam. 1. How much powder will fill the shell whose in- 
ternal diameter is 9°1 inches ? Ans. 13,2,1b nearly. 
Exam. 2. How much powder will fill a shell whose inter- 
nal diameter is 7 inches ? Ans. 6]b. 


PROBLEM VII. 


To find.how much Powder will fill a Rectangular Box. 
Finn the content of the box in inches, by multiplying the 
length, breadth, and depth all together. Then divide by 30 
for the pounds of powder. 


EXAMPLES. 


Exam. 1. Required: the quantity of powder that will fill a 
box, the length being 15 EEB ete the breadth 12, and the depth 
10 inches? © Ans. 60\b. 

Exam. 2. 


S44. POWDER AND SHELLS, &c. 


Exam. 2. How much perde vel fill a cubical box whosé 
Side is 12inches? > _ Ans. 573)b, 
PROBLEM Minig 


To find how much Powder will fill a Cylinder. 


“Mutriery the square of the diameter by the length, then 
divide by 38°2 for the pauns of powder. 


“EXAMPLES. 


Exam. 1, How much powder will the cylinder hold, whose 
mde is 10 inches, and teneth 20 inches ? 
Ans. 5211b nearly. 
_. Exam. 2. How much powder can be contained in the cy- 
linder whose diameter is 4 inches, and length 12 inches? — 
Ans. 5,3 1b. 
PROBLEM IX. 


To fi nd the Size of a, Shell to contain a Given Weight of Powder, 


Mutqriety the pounds.of powder hy 57°3, and. the cube 
root of the product will be the’ diaineter in inches. 
~ EXAMPLES. *. 
Exan. 1, What is the diameter of a shell that will hold 134 
of powder ? Ans 9:1 inches. 
Exam. 2. What is the diameter of a shell to contain 6Ib 
of powder ? - Ans.°7 inches. 
PROBLEM x. 


To find the Size bof a Cubical Box to contain a given Weigh cof 
Powder. ” 


: Mobrivey the weight in. pounds by 30, and the cube root 
of the product will be the side of the box in inches. 


EXAMPLES. » 


Exam. 1. Required the side of a cubical. box, to hold 501b 

of gunpowder? =~ Ans. 11: 44 inches. 

Exam. 2. Required the side of a cubical box, to hold 400lt 

of gunpowder ? | mee Ne Ans. 22° 89 inches. 

’ PROBLEM XI. : 
To dae what Length of a Cylinder will be filled by watt | 
ue Weight of Gunpowder. oP Saree 

- Murtiecy the weight in pounds by 38-2, and divide the 

product hs the square of the diameter in inches, for the 
length. ! : : 

EXAMPLES. 


PILING OF BALLS AND SHELLS. 248 
~ EXAMPLES. ~ 
Exam. 1. ‘What | Aepeth: of a 36- pounder gun, of 62 inches 
diameter, will be filled with 12lb of gunpowder. 
Ans. 10°314 inches, 


Exam. 2. What length of a cylinder, of 8 inches diameter, 
may be filled with 201b of powder ? Ans, 113% inches. 


OF THE PILING OF BALLS AND SHELLS. 


Iron Balls and Shells are commonly piled by horizontal 
courses, either in a pyramidical or in a wedge-like form: the 
base being either an equilateral triangle, or a square, ora 
rectangle. In the triangle and square, the pile finishes in a 
single ball; but in the rectangle, it finishes in a single row of 
balls, hike an edge. 

In triangular and square piles, the number of horizontal 
rows, or courses, is always equal to the number of balls in 
one side of the bottem row. And in rectangular piles, the 
number of rows is equal to the number of balls in the breadth 
of the bottom row. Also, the number in the top row, or edge, 
is one more than the difference between the efeneth and breadth 
of the bottom row. : 


pRDuL Est I. 
“To find the number of Balls in a Triangular Pile. 
Muttre.y continually together the number of balls in one 
side of the bottom row, and that number increased by 1, also 


the same number increased by 2; then } of the last product 
will be the answer. 





mig MEL Pe | RE et) 
‘That is, sae is the number or sum, where n is 
the number in the bottom row. | 
EXAMPLES. 


Exam. 1. Required the number of balls ina triangular pile, 
each side of the base containing 30 balls? =| Ans. 4960. 

Exam. 2. How many balls are in the triangular pile, each 

side. of the heke gpa auniog 20? . AN _ Ans. 1540. 


; PROBLEM 


246 ‘PILING OF BALLS AND SHELLS, 
PROBLEM Il. 


To find the Number of Balls in a Square Pile. : 


Muxrirty continually together the number in one side of, 
the boitom course, that number increased by 1, and double 
the same number increased Bye bs uEhee 4 of the last pene 
will be the answer. 








. That ts, SH ska . ea) is the number. 
EXAMPLES. 
Exam. 1. How many balls are ma square pile of 30 rows ? 
Ans. 9455. 
Exam, 2. How many balls are in a anuAte pile of 20 rows ? 
Ans. 2870. 


PROBLEM Ill. 
To find the Number of Balls ina Rectangular Pile. 


From 3 times the number in the length of the base row sub- 
tract one less than the breadth of the same, multiply the re- 
‘mainder by the same breadth, and the product by one more 
than the same, and divide by 6 for the answer. 

Thatis, b. heed: eetead Se 
length, and & the breadth of the lowest course. . 

Note. In all the piles the breadth of the bottom 1s equal to 
the number of courses. And jn the oblong or rectangular 
’ pile, the top row is one more than the difference between the 
length and breadth of the boston: 


is the number ; where lis the 4 


EXAMPLES. 


Exam. 1. Required the number of balls in a rectangular 
pile, the length and breadth of the base row being 46 and 15? 
Ans. 4960. 

Exam. 2, How many shot are ina rectangular complete 
an the length of the bottom course being 59, and its breadth 
Ans. 11060. 


PROBLEM IV. 
To ‘ad the Number of Balls in an Incomplete Pile. 


Front the number in the whole pile, considered as com- 
plete, subtract the number in the upper pile which is want- 


ing 


DISTANCES BY SOUND, 247 


ing at the top, both computed by the rule for their proper 
form ; and the remainder will ue the number in the frustum, 
or perce es aby 


EXAMPLES. 

Exam. 1. To find the number of shot in the incomplete 
triangular pile, one side of the bottem course being 40, and 
the top course 20 ? Ans. 10150. 

Exam. 2. How many shot are in the incomplete triangular 
pile, the side of the base being 24, and of the top 8? 

Ans. 2516. 

Exam. 3. How. many balls are in the incomplete square 
pile, the side of the base being 24, and of the top 8 ? 

Ans. 4760. 

Exam. 4. How many shot are in the incomplete rectangu- 


lar pile, of 12 courses, the length and breadth of the base 
being 40 and 20? _Ans. 6146, 


OF DISTANCES BY Su cakatiemcerg imate! OF SOUND. 


By various experiments it has been found, that sound flies, 
through the air, uniformly at the rate of about 1142 feet in 
1 second of time, or a mile in 42 or ‘4 seconds. And there- 
fore, by proportion, any distance may “be found corresponding 
to any given time; namely, multiplying the given time, in 
seconds, by 1142, for the corresponding distance in feet; or 
taking = of the given time for the distance in miles. Or di- 
viding any given distance by these numbers, to find the cor- 
responding time. 

Note. The time for the passage of sound in the interval be- 
tween seeing the flash of a gun, or lightning, and hearing the 
report, may be observed by a watch, ora small pendulum. 
Or, it may be observed by the beats of the pulse in the wrist, 
counting, on an average, about 70 to a minute for persons in 
moderate health, or 5} pulsations to a mile; and more or less 
according to circumstances. 


EXAMPLES, | 


Exam. 1. After observing a flash of of lightning, it was 12 
seconds before the thunder was heard ; required the distance 
of the cloud from whence it came ? | Ans. 24 miles. 

Exam. 2. How long, after firing the Tower guns, may the 

report 


ww 
943. PRACTICAL EXERCISES. 


report be heard ‘at Shooter’s-Hill, supposing the distance to 
be '@ miles ina straight line? Ans. 371 seconds. 
Exam. 3, After observing: the firing of a large cannon at a 
distance, it was 7 seconds before the report was heard ; what 
was its distance ? Ans. 12 mile. 
Exam. 4. Perceiving a man at a distance hewing down a 
tree with an axe, | remarked that 6 of my pulsations passed 
between seeng him strike and hearing the report of the blow ; 
what was the distance between us, allowing 70 pulses to a 
minute? - Ans. 1 mile and 198 yards. 
Exam. 5. How far off was the cloud from which thander. 
issued, whose report was 5 pulsations after the flash of light- 
ning ; counting 75 to a minute ? - Ans. 1523 yards. 
Exam. 6. If I see the flash of acannon, fired by a ship in 
distress at sea, and nee ne report 33 seconds after, how far 
is she off? Bos Ans, 7 miles, 


PRACTICAL EXERCISES IN MECHANICS, STATICS, 
-HYDROSTATICS, SOUND, MOTION, GRAVITY, PRO- 
JECTILES, AND OTHER BRANCHES OF NATURAL 
PHILOSOPHY. 


Deacitin 1. _Requirep the weight of a cast iron ball of 3. 
inches: diameter, supposing the weight of a cubic inch of the 
metal to be 0: 258lb avoirdupois ? Ans. 3°6473¢Ib. 

Quest. 2. To determine the weight. of a hollow spherical 
iron shell, 5 inches in diameter, the thickness of the metal 
being one inch ? im Pos P3:838 1b, 

Quest. a Being one day ordered to observe how far.a 
pattery of cannon was.from me, | counted, by my watch, 17 
seconds between the time of seeing the flash and hearing the 
report ; what then was the distance ? : Ans. 32 miles, 
Quest. 4, It is proposed to determine the proportional 

uantities of matter inthe earth and moon ; the density of the 
former being to that of the latter, as 10 to 7, and their diame- 
_ fers as 7930 toe 2160. _ » Ans. as 7] to 1 nearly. 
Quest. 5, What difference is there, in point of weight, 
between a block of marble, containing 1 cubic foot anda halt, 

and au of brass of the same dimensions fate 
“Ans. 496Ib (402, 

| - honey. 6. In the walls of Balbeck i in Turkey, the ancient 

Hoopolis there are three stones laid end to end, now in sight, 

that 


IN NATURAL PHILOSOPHY. 249 


that measure in length 61 yards; one of which in particular 
is 21 yards or 63 feet long, 12 feet thick, and 12 feet broad : 
now if this block be nacels, what power would balance it, so 
as to prepare it for moving ? 


Ans.. 683,% tons, the burden of an East-India ship. 
Quest. 7. The battering-ram of Vespasian weighed, sup- 
pose 10,000 pounds ; and was moved, let us admit, with such 
a velocity, by strength of hand, as to pass through 20 feet in 
one second of time; and this was found sufficient to demolish 
the walls of Jerusalem. ‘1 he question is, with what velocity 
a 32lb ball must move, to do the same execution ? 
Ans. 6250 feet. 
Quest. 8. There are two bodies, of which the one contains 
25 times the matter of the other, or is 25 times heavier ; but 
the less moves with i000 times the velocity of the greater ; : 
in what proportion then are the momenta, or forces, with which 
they moved ? 
Ans. the less moves with a force 40 times greater. 


Quest. 9. A body, weighing 20]b, is impelled by sucha 
force, as to send it through a 100 feetin a second ; with what 
velocity then would a body of 8lb weight move, if it were 
impelled by the same force ? ‘Ans. 250 feet per second. 

Quest. 10. There are two bodies, the one of which weighs 
100lb, the other 60; but the less body is impelled by a force 
8 times greater than the other; the proportion of the veloci- 
ee with which these bodies move, is required ? 

Ans. the velocity of the greater to that of the less, as 3 to 40. 


Quest. 11. There are two ‘bodies, the greater contains 8. 
times the quantity of matter in the less, and is moved with a — 
force 48 times greater ; the ratio of the velocities of these 
two bedies is required ? 

Ans. the greater is to the less, as 6 to 1. 

Quest. 12. There are two bodies, one of which moves 40 
times swifter than-the other ; but the swifter body has moved 
only one minute, whereas the other has been in motion 2 
hours: the ratio of the spaces described by these two bodies 
is required ? 

Ans. the swifter is to the slower, as 1 to 3. 

Quest. 13. Supposing one body to move 30 times swifter 
than another, as 2'so the swifter to move 12 minutes, the other 
only 1: what difference will there be between the spaces 
described by them, supposing the last has moved 5 feet? 

Ans. 1795 feet. 

Quest. 14. There are two bodies, the one of which has 
passed over 50 miles, the other only 5; and the first had 

Vou. IT. re aie moved 


250 PRACTICAL EXERCISES 


moved with 5 times the celerity of the second; what is the 
ay of the times they have been in describing those spaces ? 
‘Ans. as 2 to Ye 
Fear. 15. If alever, 40 effective inches long, will, bya 
certain power thrown successively on it, in 13 hours, raise a 
weight 104 feet ; in what time will two other levers, each 18 
effective inches long, raise an equal nad ts 73 feet ? 
Ans. 10 hours 81 minutes. 
Ques. 16. What weight will a man be able to raise, who 
presses with the force of a hundred and a half, on the end of 
an equipoised handspike, 100-inches long, meeting with a con- 
venient prop exactly 74 inches from the lower end of the ma- 
chine ? Ans. 2072Ib. 
Quest 17. A weight of 13. Ib, laid on the shoulder of aman, 
is no greater burden to him than its absolute weight, or 24 
ounces: what difference will he feel between the said weight 
applied near his elbow, at 12 inches from the shoulder, andin 
the palm of his hand, 28 inches from thesame ; and how much 
more must his muscles then draw to sapport it at right angles, 
that is, having his arm stretched right out ? 
Ans. 24lb avoirdupois. 
Quest. 18. What weight hung on at 7U inches from the 
centre of motion of a steel-yard wil? balance a small gun of 
91 cwt, freely suspended at 2 inches distance from the said 
centre on the contrary side ? Ans. 302Ib. 
Quest. 19. It is proposed to divide fe beam of a steel- 
yard, or to find the points of division where the weights of 
t, 2, 3, 4, &c. Ib, on the one side, will just balance a constant 
weight of 9alb at the distance of 2 inches on the other side 
‘of the fulcrum; the weight of the beam being’. 10bb,, and its 
whoie length 32 inches ? 
Ans. 30, 15, 10, 73, 6, 5, 42, 32, 31, 3, 2%, 2t, &e. 
Quest. 20. ‘Two men carrying a burden of 200Ib weight 
between them, hung on a pole, the ends of which rest on their 
shoulders ; how much of this load is borne by each man, the 
weight hanging 6 inches from the middle, and the whole length 
of the pole being 4 feet ? ~~ Ans. 125lb and 75lb. 
Quest. 21. if, in a pair of scales, a body weigh 90Ib in 
one scale, and only 40}b in the other ; required its true weight, 
and the proportion of the lengths of the two arms of the ‘ba- 
lance beam, on each side of the point of suspension ? 
Ans. the weight 601b. and the proportion 3 to 2. 
Quest. 22. To find the weight of a beam of timber, or 
other body, by means of man’s own weight. or any other 
weight. For instance, a piece of tapering timber, 24 feet 
Jong, being laid over a prop, or the edge of another beam, 
| 1s 


IN NATURAL PHILOSOPRY. 251 


is found to balance itself when the prop is 13 feet from the 
tess end ; but Temoving the prop a foot nearer to the said end, 
it takes a man’s weight of 210]b, standing on the Jess end, te 
held it in equilibrium. Required the weight of the tree ? 
Ans. 2520\b. 
Quesr. 23. If an be a cane or walking-stick, 40 inches 
long, suspended by a string sp fastened to the middie point p : 
now a body being hung on at £, 6 inches distance from p, is 
balanced by a weight of 2lb, hung on at the larger end a ; but 
removing the body to Fr, one inch nearer to D, the 2lb weight 
on the other side is moved to c, within 8 inches of p, before 
the cane will restin equilibrio, Required the weight of the 
body ? Ans. 24\b. 
Quest. 24. If an, pc be two inclined planes, of the lengths 
of 30 and 40 inches, and moveable about the joint at B ; what 
will be the ratio of two weights p, a, in equilibrio on the planes, 
in all positions of them : and what will be the altitude sp of 
the angle B above the horizontal plane ac, when this is 50 
inches long ? 
Ans. Bp=24; and Pp to q as aB to Be, or as 3 to 4. 
Quest. 25. A lever, of 6 feet long, is fixed at right angles 
in a screw, whose threads are | inch asunder, so that the 
lever turns just onee round in raising or depressing the screw 
one inch. If then this lever be urged by a weight or force 
of 50lb, with what force will the screw press ? 
_ Ans. 226192Ib. 
Quest. 26. If aman can draw a weight of 1501b. up the 
side of a perpendicular wall, of 20 feet high; what weight 
will he be able to raise along a smooth plank of 30 feet long, 
laid aslope from the top of the wall ? Ans. 225lb. 
Quest. 27. lf a force of 150lb be applied on the head ef 
arectangular wedge, its thickness being 2 inches, and the 
length of its side 12 inches ; what weight will it raise or ba- 
lance perpendicular to its side ? Ans. 900Ib. 
Quest. 28. If around pillar of 30 feet diameter be raised 
on a plane inclined to the horizon in an angle of 75°, or the 
shaft inclining 15 degrees out of the perpendicular : what 
length will it bear before it overset ? 
Ans. 30-(2-++,/3)*or 111-9615 feet. 
Quest. 29. If the greatest angle at which a bank of natu- 
ral earth will stand be 45° ; it is proposed to determine what 
thickness an upright wall of stone must be made throughout, 
just to support a bank of 12 feet high; the specific gravity of 
the stone being to that of earth, as 5 to 4. 
‘Ans, 43/1, or 429325 feet. 
QUEST. 30. If aha: stene wall be made Tike a claire or 
a ing 


252 PRACTICAL EXERCISES 


having its upright section a triangle, tapering to a point at 
top, but its side next the bank of earth perpendicular to the 
_ horizon ; what is its thickness at the bottom, so as to support 
the same bank ? Ans. 12,/2 or 5:36656 feet. 
Quest. 31. But if the earth will only stand at an angle of 
30 degrees to the horizontal line ; it is required to determine 
the thickness of wall in both the preceding cases ? 
_ Ans. the breadth of the rectangle 124/14, or 5:36656, 
but the base of the triangular bank 12,78, or 6°53667. 


Quest. 32. To find the thickness of an upright rectangu- 
lar wall. necessary to support. a body of water; the water 
being 10 feet deep, and the wall 12 feet high ; also the spe- 
cific gravity of the wall to that of the water as I to %, 

Ans. 4°204374 feet. 

Quest. 33. To determine the thickness of the wall at the 
bottom, when the section of it is triangular, and the altitudes 
as before. Ans. 5°1492866 feet. 

Quest. 34. Supposing the distance of the pier from the 
sun to be 95 millions of miles ; 1 would know at what distance 
from him another body must be placed, so as to receive light 
and heat quadruple to that of the earth ? 

Ans. at half the distante, or 472 millions. 

Quest. 35. If the mean distance of the sun from us be 
106 of his diameters ; how much hotter is it at the surface of 
the sun, than under our equator ? 

. Ans. 11236 times hotter. 

Quest. 36. The distance between the earth and the sun 
being accounted 95 millions of miles, and between Jupiter and 
the sun 495 millions ; the degree of light and heat received 
by Jupiter, compared with that of the earth, is required ? 

Ans. Zo', or nearly ;', of the earth’s s light and heat. 


 Quusr.. 37. A certain body on the surface of the earth 
weighs acwt. or 112Ib; the question is whither this body 
must be carried, that it may weigh only 10]b ? 
Ans. either at 3°3466 semi-diameters, or -$; of 
a semi-diameter, from the centre. 
Quest. 38. If a body weigh 1 pound, or 16 ounces, on the 
surface of the earth; what will its weight be at 50 miles 
-above it, taking the earth’s diameter at 7920 miles ? 
: Ans. 150z. 93dr. nearly. 
Quest. 39. Whereabouts, in the line between the earth 
and moon, is their common centre of gravity ; supposing the 
earth’s diameter to be 7920 miles, and the moon’s 2160; also 


the 


IN NATURAL PHILOSOPHY. 253: 
the density of the former to that of the latter, as 99 to 68, or 
as 10 to 7 nearly, and their mean distance 30 of the earth’s 
diameters ?.Ans. 633°65 miles below the surface of the earth. 


Quest. 40. Whereabouts, between the earth and moon, 
are their attractions equal to each other? Or where must 
another body be placed, so as to remain suspended in equi- 
librio, not being more attracted to the one than to the other or 
having no tendency to fall either way ? their dimensions being 
as in the last question. 

Ans. From the earth’s centre 26:8 ? of the earth’s 
From the moon’s centre 3:2 diameters. 


Quest. 41. Suppose a stone drept into an abyss, should be 
stopped at the end of the 11th second after its delivery ; what 
space would it have gone through ? Ans. 1946, feet. 


Quest. 42. What is the difference between the depths of 
two wells, into each of which should a stone be dropped at 
the same instant, the one will strike the bottom at 6 seconds 
the other at 10? Ans. 10292 feet. 


Quest. 43. If a stone be 19} seconds in descending from 
the top of a precipice to the bottom, what is its height ? 
: Ans. 611511 feet. 


Quest. 44. In what time will a musket ball, dropped from 
the top of Salisbury steeple, said to be 400 feet high, reach 
the bottom ? __ Ans. 5 seconds nearly. 


Quest. 46. If a heavy body be observed to fall through 
100 feet in the last second of time, from what height did it 
fall, and how long was it in motion ? | 

td Ans. time 3228 sec. and height 2094273 feet. 


Quesr. 46. A stone being let fall into a well, it was observ- 
ed that, after being dropped, it was 10 seconds before the 
sound of the fall at the bottom reached the ear. What is the 
depth of the well? Ans. 1270 feet nearly. 


Quest: 47. It is proposed to determine the length of a 
pendulum vibrating seconds, in the latitude of London, where 
a heavy body falls through 16,1, feetin the first second of time ? 

. Ans. 39:11 inches. 

By experiment this length is found to be 39} inches: 


Quest. 48. 


254 PRACTICAL EXERCISES 


Quest. 48. What is the-leagth of a pendulum vibrating in 
2 seconds: also in half a second, and in a quarter second ? a 
Ans. the 2 second pendulum 1561 ni 
_ the £ second pendulum 25 
the }. second pendulum 22,4 inches. 


_ Quest. 49. What diflerence will there be in the number of 
vibrations, made by a pendalum of 6 inches long, and another 
of 12 inches long, in anhour’stime? Ans. 26923, 


Quest. 50. Observed that while a stone was descending, 
to measure the depth of a well, a string and plummet, that 
from the point of suspension, or the place where it was held, 
to the centre of oscillation, measured just 18 inches, had made 
8 vibrations, when the Eound from the bottom returned. What 
was the depth of the well? _ Ans. 412°61 feet. 


Quest. 51. If a ball vibrate in the arch of a circle, 10 de- 
grees on each side of the perpendicular ; or_a ball roll down 
the lowest 10 degrees of tie arch ; required the velocity at 
the lowest point? the radius of the circle, or length of the 
pendulum, being 20 feet. Ans. 4-4213 feet per second. 


Quest. 52. If a ball descend down a smooth inclined 
plane, whose length is 100 feet, and altitude 10 feet ; how 
jong will it be in descending, and what will be the last velo- 
city? 

Ans. the veloc. 25:364 feet per sec. and time 7:8352 sec. 


Quest. 53. If a cannon bail, of ilb weight, be fired against 

a pendulous block of wood, and striking the centre of oscilla- 

tion, cause it to vibrate an arc whose chord is 30 inches ; the 

radius of that arc, or distance from the axis to the lowest point 

ot the pendulum being 118 inches, and the pendulum vibrating 

in smal arcs 40 oscillations per minute. Required the velo- 

city of the ball, and the velocity of the centre of oscillation 

of the pendulum, at the lowest point of the arc; the whole 
weight of the pendulum being 500th ? 

Ans. veloc. ball 1956*6054 feet per sec. 

and veloc. cent. oscil. 3°9054 feet per sec. 


Quest. 54, How deep will a cube of oak sink in common 
water ; ; each side of the cube being 1 fuot ? 
Ans. 11,5 inches. 


Quest. 55. How deep will a globe of oak sink in water ; 
the diameter being 1 foot ? Ans. §$°9867 inches. 
, QUEST. 


iN NATURAL PHILOSOPRY. 256 


Quest. 56. If a cube of wood, floating in common water, 
have three inches of it dry above the water, and 4,2. inches 
dry when in sea Water ; it is proposed to determine the mag- 
nitude of the cube, .and what sort of wood it is made of ? 

_ Ans. the wood 1s.oak, and each side 40 inches. 


Quest. 87. An irregular piece of lead ore weighs, in air 
~ £2 ounces, but in water only 7; and another fragment weighs 
in air 142 ounces, but in water only 9; required their com- 
parative densities, or specific gravities ? 

Ans. as 145 to 132. 


- Quest. 58. Anirregular fragment of glass, in the scale, 

weighs 171 grains, and another of magnet 102 grains; bat in 

water the first fetches up no more than 120 grains, and the 

other 79: what then will their specitic gravities turn out to 

be ? Ans. glass to magnet as 3933 to 5202, 
or nearly as 10 to 13. 


Quest. 59. Hiero, king of Sicily, ordered his jeweller to | 


make him a crown, contaming 63 ounces of gold. ‘The work- 
men thought that substituting part silver was only a proper 
perquisite ; which taking air, Archimedes was appointed to 
examine it; who on putting it into a vessel of water, found 


it raised the fluid 8°2245 cubic inches: and having discover- 


ed that the inch of gold more critically weighed 10°36 ounces, 
and that of silver but 5°85 ounces, he found by calculation 
what part of the king’s gold had been changed. And you are 
desired to repeat the process. 
Ans. 28-8 ounces. 


Quest. 60. Supposing the cubic inch of .common glass 
weigh 1°4921 ounces troy, the same of sea-water °59542, and 
of brandy 5368 ; then a seaman having a gallon of this li- 
quor in a glass bottle, which weighs 3-84lb out of water, and, 
to conceal it from the officers cf the customs, throws it over- 
board. It is proposed to determine, if it will sink, how much 
force will just buoy it up? Ans. 14-1496 ounces. 


Quest. 61. Another person has half an anker of brandy 
of the same specific gravity as in the last question ; the wood 
of the cask suppose measures 1 of a cubic foot ; it is propos- 
ed to assign what quantity of lead is just requisite to keep the 

cask and liquor under water ? Ans. 89°743 ounces. 


Quest. 62. Suppose, by measurement, it be found that a 
man-of-war, with its ordimance, rigging, and appointments, 
sinks 


256 PRACTICAL EXERCISES | 
sinks so deep. ag Ft displace 50000 cubic feet of fresh. water i 
what i is the whole weight of the vessel ? 

Ans. 13952, , tons. 


Quest. 63. It is ronjuired to determine what would be the 
height of the atmosphere, if it were every where of the 
same density as at the surface of the earth, when the quick- 
silver i in the barometer stands at 30 inches ; and also. what 
would be the height of a water barometer at the same time ? 
: Ans. height of the air 291662 feet, or 5°5240 miles, 
: - height of water 35 feet. ‘ 
ae 
Quest. 64. With what velocity would each of those three 


-. fluids, viz. quicksilver, water, and air, issue through a small 


orifice in the bottom of vessels, of the respective heights of 
30 iuches, 35 feet, and 5°5240 miles, estimating the pressure 
by the mnie altitudes, and the air rushing into a vacuum ? 
Ans. the veloc. of quicksilver 12-681 feet. 
the veloc. of water - 47°447 
the veloc. of air - - 1369°8 


Quest. 65. A very large vessel of 10 feet high (no matter 
what shape) being kept constantly full of water, by a large 
supplying cock at the top ; if 9 small circular holes, each } 
of an inch diameter, be opened i in its perpendicular side at 
every foot of the depth: It is required to determine the se- 
veral distances to which they will spout on the horizontal 
plane of the base, and the quantity of water discharged by all 
of them in 10 minutes ; 


‘Ans. the distances are 
4/36 or 6:00000 
4/64 - 8-00060 
4/84 - 9:16515 
JfI96 - 9°79'796 
4/100 - 10:00000 
/96 - 9°79796 
4/84 - 916515 
4/64 - 880000 
4/36 - 6-06000 | 
and the quantity discharged in 10 min. 123-8849 gallons. 
Note. In this solution, the velocity of the water is supposed 
to be equal to that which is acquired by a heavy body in fall- 
ing through the whole height of the water above the orifice, 
and that it is the same in every part of the holes. 
‘ QUEST. 


IN NATURAL PHILOSOPHY. 257 


Quest. 66. If the inner axis of a hollow wube of copper, 
exhausted of air, be 100 feet ; what. thickness must. it be of, 


that it may hae float in “ air ? 
Ans 02688 of cok thick. 


Quran: 67. i a mphotical balloon of. copper, of ; i; of an 
inch thick, have its cavity of 100 feet diameter, and be filled 
with inflammable air, of 3, of the gravity of common air, 
what weight will just balance it, and a it from rising up 


~ into the atmesphere ? ta a. . _ Ans. 21273Ib. 


Quest. 68. If a glass tube, 36 inches ae close ‘at top be 
sunk perpendicularly into water, till its lower or open. end be 
30 inches below the surface of the water; how high will the 
water rise within the tube, the quicksilver i in the common ba- 


rometer at the same time standing at 291 inches ? 
“Ans. 226545 inches. 


Quest. 69. If a diving bell, of the form of a parabolic cox 
noid, be let down into the sea to the several depths of 5, 10, 
15, and 20 fathoms ; it is required to assign the respective 


: heights to which the water will rise within it ; its axis and the 


diameter of its base being each 8 feet, and the quicksilver i in 
the barometer standing at 30:9 inches ? y 
Ans. at 5 fathoms deep the water rises 2:03546 feet, 


at tO, ste - 3-06393 
RET oe vel es ee  SUOSET 
MRO ee te es 4:14653 


my 


Vault? eee reagaed ae Se ON 


[ 268}. 


“ON THE NATURE AND SOLUTION OF EQUATIONS 
IN RAL. 


oh In order to investigate: due general properties of the 
higher equations, let there be assumed between an unknown 
quantity x, and given quantities a, 5, c, d, an equation consti- 
tuted of the continued product of uniform factors : thus 

(wa) X (w—b) X (xc) X (a—d) = 0. 

This, by performing rh multiplications, and arranging the 
final product according to the powers or dimensions of x, be- 
comes . 
gta )x+ab \2*—abe ) x + abed=0.... (A) 


—b +-ac —abd 
=e +ad —acd 
—a +be —bed 
 +bd 
-+-cd 


Now it is obvious that the sevomiblane of terms which compose 
the first side of this equation may become equal to nothing i in 
four different ways; namely, by supposing either x =a, or 
“x=b,orx=c, oraz =d; for in either case one or other 
of the factors x—a, x— oh x—¢, s—d, will be equal to no- 
thing, and nothing multiplied by any quantity whatever will 
give nothing for the product. If any other value e be put 
Sar x, then none of the factors e—a, e—b, e—c, e—d, being 
equal to nothing, their continued product cannot be equal to 
nothing. There are therefore, in the proposed equation, four 
roots or values of x ; and that which characterizes these roots 
is, that on substituting each of them successively instead of x, 
the aggregate of the terms of he saaees Niemann by.the 
opposition of the signs +- and—. 

The preceding equation is only of the fourth power or de- 
gree ; but it is manifest that the above remark applies to equa- 
tions of higher or lower dimensions: viz. that in general an 
equation of any degree whatever has as many roots as there 
‘are units in the exponent of the highest power of the un- 
known quantity, and that each root has the property of ren- 
dering, by its substitution in place of the unknown quantity, 
the aggregate of all the terms of the equation equal to no- 
thing. 

-. It must be bisenved that we cannot have all at once x = a, 
c= 6, +=c, &c. for the roots of the equation ; but that the 
particular equations e—-a=0,2-b=0,21 —c=0, &e. 
obtain only ina disjunctive sense. They exist as factors in 

| the 


EQUATIONS. 259 - 


the game equation, because algebra gives, by one and the same 
formula, not only the solution of the particular problem from 
which that formula may have originated, but also the solution 
of all problems which have similar conditions. The different 
roots of the equation satisfy the respective conditions : and 
those roots may differ from ene another, by their quantit y 
and by their mode of existence. 

It is true, we say frequently that the roots of an equation 
arex =a, 24 = 6,2 = Cc, &c. as though those values of x 
existed conjunctively ; ; but this manner of speaking is an ab- 
breyiation, which it is necessary to understand in the sense 
explained above. 

2. In the equation a all the roots are positive ; but if the 
factors which constitute the equation had been x ne a,x + b, 
x +c, x + d, the roots would have been negative or sub- 
tractive. Thus 

z4-+-a)jx? +ab>\22 -+abe e-eabed0. aro de) 

+-b +ae +abd 
+e --ad ctacd 
+d be bbed 


‘ -+-bd 
+cd 
has negative roots, those roots beng += —a,7 = — 6b, 
2£=—c,xc = — d: and here again we are to apply them 
disjunctively. 


3, Some. equations have their roots in part noniive: in part 
negative. Such is the seltowpe: 


a3—a) 22-+ab) x+abe=0. 2 wk (C). 
—b>5  —ac 
+e —be ‘ 


Here are the two positive roots, viz. x =a, a2 = 6; and one 
negative root, viz. = — ¢: the equation being constituted 
of the continued product of the three factors, | r—a=0, cmb 
=0, r-+c=0. 

From an inspection of the equations A, B, ¢, it may be in- 
ferred, that a complete equation consists of a number of terms 
exceeding by unity the number of its roots. 

4. The preceding equations have been considered as form- 
ed from equations of the first degree, and then each of them 
contains so many of those constituent equations as there are 
units in the exponent of its degree. But an equation which 
exceeds the second dimension, may be considered as composed 
of one or more equations of the second degree, or of the 
third, &c. combined, if it be necessary, with equations of the 
first degree, in such manner, that the product of all those 


constituent equations shall form the proposed equation. ‘ a 
eed, 


260 - EQUATIONS. 


deed, when an equation is formed by the successive multipli- 
cation of several simple equations, quadratic equations, cubic 
equations, &c. are formed ; which of course may be regent: 
ed ‘i factors of the resulting equation. 

. It sometimes happens that an equation contains imagina- 
Try oot: and then they will be found also in its constituent 
equations. ‘T’his class of roots always enters an equation by 
pairs ; because they may be considered as containing, in their 
expression at least, one even radical place before a negative 
quantity, and because an even radical is necsssarily preceded 
by the double siga +. Let, for example, the equation be x4 
—(2a—2c) «3+ (a? +b2 —4ac-+-c? +d?) x24 (2a? ¢+-2b%¢ — 
Zac? —2ad?)x+-(a?+4-b2). (c?-+-d?) =0. This may be re- 
garded as constituted of the two subjoined quadratic equa- 
tions, 22 —2ax-+-a? +b? = 0, x? ++2cx-+-c -+-d?=0: and each 
of these quadratic contains two imaginary roots; oe first 
giving 7=a +b), and the second s=—citd 

In the equation resulting from the product of as two 
quadratics, the coefficients of the’ powers of the unknown 
quantity, and of the last term of the equation, are real quan- 
tities, though the constituent equations contain imaginary 
quantities ; the reason is, that these latter disappear by means 
of addition and multiplication. 

The same will take place in the equation (x—a) . (x+) . 
(x? -+-2cx-+-c?-+-d?) = 0, which is formed of two equations of 
the first degree, and one equation of the second whose roots 
are imaginary. 

‘These. remarks being premised, the subsequent general 
theorems will be easily established. 


THEOREM 5 i 


Whatever be the Species of the Roots of an Equation, when 
the Equation is arranged according to the Powers of the 
Unknown quantity, if the First Term be positive, and have 
unity for its Coefficient, the following: Properties. may. be 
traced : 

I. The first term of the equation is the unknown quantity 
raised to the power denoted by the number of roots. 

Il. The second term contains the unknown quantity raised 
to a power less than the former by unity, with a coefficient 
equal to the sum of the roots taken with contrary signs. 

III. The third term contains the unknown quantity raised 
to a power less by 2 than that of the first term, with a coeffi- 
cient equal to the sum of all the products which can be form- 
ed by multiplying all the roots two and two. 

IV. 


EQUATIONS. 261 

iV. The fourth term contains the unknown quantity raised 
to a power less by 3 than that.of the first term with a coeffi- 
cient equal to the sum of all the products which can be made 
by multiplying any three of the roots with contrary signs. 

V.. And so on to the last term, which is the continued pro- 
duct of all the roots taken with contrary signs. | 

All this is evident from inspection of the equations exhibit- 
ed in arts. 1, 2,.3, 5. 

Cor. 1. Therefore an equation having all its roots real, 
but some positive, the others negative, will want its second 
term when the sum of the positive roots is equal to the sum 
of the negative roots. .Thus, for example, the equation c 
will want its second term, if a ee b=.¢, 

Cor. 2. An equation whose roots are all imaginary, will 
want the second term, if the sum of the real quantities which 
_ enter into the expression of the roots, is partly positive, partly 
_ negative, and has the result reduced to nothing, the 1 imagina- 
ry parts mutually destroying each other by addition in each 
pair of roots. Thus, the first equation of art. 5 will want. 
the second term if — 2a + 2¢=0,ora=c. The: second 
equation of the same article, which’ has its roots partly real, 
partly imaginary, will want the second term if § — a F go == 0, 
or a—b = 2c, 

Cor. 3. An equation will tan its third term, if the sum of 
the products of the roots taken two and two, is partly positive, 
partly negative, and these mutually destroy each other. 

Remark. An incomplete equation may be thrown into the 
form of complete equations, by introducing, with the coeflicient 
a cypher, the absent powers of the unknown quantity: thus, 
for the equation «3 --+ r= 0, may be written x? + 02? + 0 
x -+r=0. .This in some cases will be useful. 

Cor. 4. An equation with positive roots may be transformed 
into another which shall have negstive roots of the same ya- 
lue, and reciprocally. In order to this, it is only necessary to 
change the signs of the alternate terms, beginning with the se- 
cond. Thus, for example, if instead of the equation 23 —8x? 
+ 17a—10 = 0, which has three positive roots 1, 2, and 5, 
we write x3 4822 ++ 1724-10 =0, this latter equation will have 
three negative rootsz =—1,7=-~2,c=—5. In like man- 
ner, if instead of the equation ge 3 Ve2 132: + 10 =0, which 
has two positive roots z = 1, «==2, and .one negative root 
x =—5, there be taken oP Qu2 — 130 — 10 = 0, this latter 
equation will have two negative roots, a, x= —2, and 
one positive root 2==5. 

In general, if there be taken the two equations, (a—a) X 
( 2—b) X X(x—e) X (aed) X &c. = 0, and (xa) X (a+b) X 

(e+e) 


962 EQUATIONS. 


(a-+e) X(x--d) X &c. = 0, of which the roots are the same 
in magnitude, but with different signs : if these equations be 
developed by actual multiplication, and the terms arranged 
according to the powers of a, as inarts. 1, 2; it will be seen 
that the second terms of the two equations will be affected 
with different signs, the third terms with like signs, the fourth 
terms with different signs, &c. 

‘When an equation has not all its terms, the deficient terms. 
must be supplied a cyphers, before the pene rule can be 


applied. 


Cor. 5. The sum of the roots of an épeatigns the sum of 
- their squares, the sum of their cubes, &c. may be found with- 
out knowing the roots themselves. . For, let an equation of 
any degree or dimension, m, be «™” -+- fa™—1 -- ga™—* -+- ha™—3 
+ &c. = 0, its roots being a, b, c,d, &c. Then we shall have, 


ist. The sum of the first powers of the roots, that is, of the 
voots themselves, or a--b--c--&e, =~—f; since the coefficient 
of the unknown quantity in the second term, is equal to the 
sum of the roots taken with different signs. 


2dly. The sum of the squares of the roots, is equal to the 
square of the coeflicient of the second term made less by twice 
the coefficient of the third term: viz. a2--b2-+-c2-++- &c. =f? 
==2¢. For, if the polynomial a+b-++c-++ &c. be squared, it 
will be found that the square contains the sum of the squares 
of the terms a, 6, c, &c. plus twice the sum of the products 
formed by multiplying two and two all the roots a, 6, c, &c. 
That is, (a--b--c-+-&c. )? == a? --b? +-c? + &e. 4-2 (ab-bac+ 
bc+&c. :), But it is obvious, from equa. a, B, that (a -- b+ ¢ 
+ &c.)? = f2, and (ab + ac + bce-+ &c. Jag. Thus we have 
f? =(a? +62 +c? L&c. Me ¢g: and a a ial a? +6? 4c 
+&c.= =f?—2¢. 


 3dly the sum of the cubes of the roots, is’ equal to 3 times 
the rectangle of the coefficient of the second and third terms 
made less by the cube of the coefficient of the second term, 
and 3 times the coefficient, of the fourth term: viz. a? + bs 
4c?-+-&¢c, =—f%+3f¢—3h. For we shall by actual involu- 
tion have eae )3==a3+-b3-+¢3 4-&c.4-3(a+b-+c) X 
(ab+-be-+-ac) —3abe. But (a+b+c+&c.)?=—/f?, (a-+-b+e 
4+-&c.) X (ab+-ace+be-+ &c.)=—fg,abe=—h. Hence there- 
fore, —f?=a?-+b?-+c3-+ &c. — 3fg-+3h ; and consequently, 

ae +b3+-c3-+&e. = — f? + 3f/g — 3h. “And so on, for other 


powers of the roots. 
"THEOREM 


EQUATIONS. 26 


on] 


THEOREM tl. 


In Beaiy Benudice, which contain only Real Races 

I. If all the roots are positive, the terms of the equation 
will be + and — alternately. 

Il. If all the roots are negative, all the terms will have the 
sign «+, 

Ill. If the roots are partly positive, partly. negative, 
there will be as many positive roots as there are variations of 
signs, and as many negative roots as there are permanencies of 
signs ; these variations and permanences being observed from 
one term to the following through the whole extent of the 
equation. 

In all these, either the equations are complete in their terms, 
or they are made so. * 

The first part of this theorem is evident from the examina- 
tion of equatien a; and the second from equation s. 

To demonstrate the third, we revert to the equation c (art. 
3), which has two positive roots, and onenegative. It thay 
happen that either c>a-+d, or c<a-+b. 

In the first case, the second term is positive, and the third 
is negative; because, having c>a-+6, we shall have ac-+-be > 
abu) Sab. And, as the last term is positive, we see that 

rom the first to the second there is a permanence of signs ; 
from the second to the third a variation of signs ; and from 
the third to the fourth another variation of signs. Thus there 
are-two variations and one permanence of signs ; ; that is, as 
many variations as there are positive roots, and as many per- 
manences as there negative roots. 

In the second case, the second term of the equation i is ne- 
gative, and the third may be either positive or negative. If 
that term is positive, there will be from the first to the second 
a variation of signs ; from the second to the third anothes 
variation ; from the third to the fourth a permanence ; mak- 
ing in all two variations and one permanence of signs. If the 
third term be negative ; there will be one variation of signs 
from the first to the second; one permanence from the second 
to the third; and one variation from the third to the fourth : 
thus making again two variations and one permanence. The 
number of variations of ‘signs’ therefore in this case, as well 
as in the former, is the same as that of the positive roots ; 
and the number of permanencies, the same as that of the ne- 
gative roots. 

Corol. Whence it follows, that if it be known, by any means 
whatever, that an equation contains tay real roots, it is also 

known 


pantie 


264 | EQUATIONS. 


known how many of them are positive, and how many nega- 
tive. Suppose, for cx it be known that, in the equa- 
tion 25 -+-3a4 — 23x23 —27x2-+1662—-120=0, all the roots are 
real: it may immediately be concluded that there are three 
positive and two negative roots. In fact this equation has the 
three positive roots s = 1, « = 2,2 =3; and two ee 
roots, z=—4, c=— 5. 

If the equation were incomplete, the aieent terms must be 
supplied by adopting cyphers for coefficients, and those terms 
must be marked with the ambiguous sign +. Thus, if the 
equation were 

LF mm QOex3 30x? =p. 192— 30=0, 
all the roots Gane real; and the second term wanting. It 
must be written thus : 
gi tOr4 — 20x23 4-302? ++192-—30=0, 
Then it will be seen, that, whether the second term be posi- 
tive or negative, there. will be 3 variations and 2 permanen- 
cies of signs: and consequently the equation has 3 positive 
and 2 negative roots. The roots in fact are, 1, 2,3,—1,—5. 

This rule only obtains with regard to equations whose roots 
are real. If, for example, it were inferred that, because the 
equation x? 4 2x2 + 5 = 0 had two permanencies of signs, it 
had two negative roots, the conclusion would be erroneous : 
for both thes roots of this equation are modgitary 


_ ‘THEOREM III. 


Every Aegan may-be Transformed. into Another whose 
Roots shall be Greater or Less by a Given Quantity. 

In any equation whatever, of which « is unknown, (the 
equations A, B, c, for example) make a=z-+-m, z being a new 
unknown quantity, m any given quantity, positive or negative ; 
then substituting, instead of a and its powers, their values re- 
sulting from the hypothesis that « =z -+-m ; so shall there 
arise an equation, whose roots shall be greater or less than 
the roots of the primitive equation, by the assumed quantity m. 

_ Corol,. The principal use of this transformation, is to take 
away any term out of an equation. Thus, to transform an 
equation into one which: shall want the second term, let m be 


a 
so assumed that nm— a= 0, or m= -, nt being the index of 
mn 


the highest power of the unknown quantity, and a the coefi~ 
cient of the second term'of the equation, with its sign chang- 
ed: then, if the roots of the transformed equation can be 
found, the roots the. original eqanton: may sees be found, 


because et +- — ssh 
THEOREM 


eaten te, 


_ EQUATIONS. 265 


THEOREM: Iv. 


Every Equation may be Transformed into Another, whose 
Roots shall be Equal to the Roots of the First Multiplied or 
Divided by a Given Quantity. 


. 1. Let the equation be 23 + az?-+ bz-++-c=0: if we put 
fe=t, or z= ti the transformed equation will be a3 fas? + 


f2bx-+-f2c=0, of which the roots are the respective products 
of the roots of the primitive equation multiplied into the quan- 
tity f. 

By means of this transformation, an equation with fractional 


quantities, may be changed into another which shall be free 
2 
from them. Suppose the equation were z*-+- ot he 


=0: multiplying the whole by the product of the denomina- 
tors, there would arise ghkz*--hkaz? +-gkbz+ghd=0 : then 


assuming ghkz=a, or z= a the transformed equa. would be 


£3 -+-hkax? -+-g2k*hba-+-g%k?h3d=0. | 

The same transformation may be adopted, to exterminate 
the radical quantities which affect certain terms of an equa- 
tion. Thus, let there be given the equation z?--az?,/k-- 
bz-+-ceyk: make z,/k=< ; then will the transformed equation 
be x3--akx3-+-bka-+-ck3=0, in which there are no radical 
quantities. . 

2. Take, for one more example, the equation z?-+-az?-- 


bz--c=0. Make “=n; then will the equation be transform- 


RC & 
ed to 23-+ suse li tale 0, in which the roots are equal to 


the quotients of those of the primitive equations divided by /. 

It is obvious that, by analogous methods, an equation may 
be transformed into another, the roots of which shall be to 
those of the proposed equation, in any required ratio. But 
the subject need not be enlarged on here. The preceding 
succinct view will suffice for the usual purposes, so far as re- 
lates to the nature and chief properties of equations. We shall 
therefore conclude this chapter with a summary of the most 
useful rules for the solution of equations of different degrees, 
besides those already given in the first volume. 

Vor. IL. me ‘35 1. Rules 


266 SOLUTION OF EQUATIONS BY SINES, &c. 


1. Rules for the Solution of Quadratics by Tables of Sines and 


Tangents. 

1. If the equation be of the form x?-+-px=q : 
Make tan a = o/s then will the two roots be, 
x==-Ftan oye: 4. ees eco 4AS]- 

2. For quadratics of the fore. C2 a NL=]. 

Make, as before, tan a= =: then will 


a ee tan BAS sk 4s Tent LAd/q 
3. For quadratics 8 the form x?--px=—q. 


Make sin a= “V4 then will 


 wsetan tag... . . xs=—cot fA /f9- 
4. For quadratics et the form «?-—pz=—q. 


Make sin a = ery then will 
o==-tan BAG wh oo 6 be x=--cot Lay/q. 
ope paeik 3 
In the last two cases, if —,/q exceed unity, sin 4 is imaginary, 


and consequently the values of x. 
The logarithmic application of these formule is very sim- 
ple. Thus, in case ist. Find a by making 
~ 10-++-log 2--1 log g—log p=log tan a. 
+ log tan 44-+1 log g—10. 
Oye ROR Re — (log cot Ia+4 log if 10).. 
' Note. This method of solving quadratics, is chiefly of use 


when the quantities p and q are large integers, or ortade 
fractions. 3 


Il, Rules for the Solution of Cubic Beiasson by tables of Sines, 
Tangents and Secants. 


1. For cubics of the form 2° Diane 
Make tan pai 2/ip - . tan a=2/tan 4B. 


Then x== cot oe 2/ip. 
2. For cubics of the form x3 —prtq=0. 


Make sin B= ae 27 ip. ss _. tan a = 2/tan is. 


Then a= = cosec 2a . 2,/ip 
Here, if the value of sins should exceed aise s would be 
imaginary, and the equation would fall in what is called 
the irreducible case of cubics. In hat case we must make 
3 - cosec 


SOLUTION OF EQUATIONS BY SINES, &c. 267 


cosec 3a ii 8 2/ip: and then the three roots would be 
c= + sin A. 24f/ip. 
a= £ sin (60°—a). 2,/1p. 
. ec sin (60°-+-a) . 2./1p. 
If the value of sin B were 1, we should have p==90°, tan 


ee 90° 
3a=1; therefore a= aes and then 


x= + sin 30°. 2,/1p=+ ip : 
a= + sin 30°.2y/ip=t sf 1p. 
x= = sin 90° .24/1p = =F Qs 1p. 


Here it is obvious that the first two roots are equal, that their 
sum is equal to the third with a contrary sign, and that this 
third is the one which is produced from the first solution*. 

In these solutions, the double signs in the value of x, re- 
late to the double signs in the value of gy. 

N. B. Cardan’s rule for the solution of Cubics is given in 
the first volume of this course. 

. Hl. Solution of Biquadratie Equations. 

Let the proposed biquadratic be 24-++-2px23==qx?-+-ra-+-s. 
Now (x?--pa-+n)? =x4-+2px3+-(p?-+-2n) 2? -+2pnx--n? : if 
therefore (p?-+-2n)x? -+-2pnx-+n?2 be added to both sides of the 
proposed biquadratic, the first will become a complete square 
(a?--px-+n)?, and the latter part (p?--2n-+q) «2-+-(2pn-+r) 
x--n?--s, is a complete square if 4( p?-+4-2n-+-q) . (n?++-s)= 
(2pn-+r)? ; that is, multiplying and arranging the terms accord- 
ing to the dimensions of n, if 8n3-+ 4qn*+- (8s— 4rp).n-+-4qs 
--4p?s—r?=0. From this equation let a value of n be ob- 
tained, and substituted in the equation (x? 4+-pxr-+n)? =(p? -+- 
2n-+¢q) x? +-(2pn-+r)ae-+-n? +s; then extracting the square 
root on both sides. 








# The tables of sines, tangents, &c. besides their use in trigonometry, and in the 
solution of the equations, are also very useful in finding the value of algebraic ex- 
pressions where extraction of roots would be otherwise required. Thus, if a and 6 
be any two quantities, of which a is the greater. Find 2, z, &c. so, that tan x =s 


a ra) ; % , 
—, sin 2=3 af —, Sec y= —, tanu == — and sin ¢ =-—: then will 
wars V/s y b? a ! a 


log ./ (a? —b?) slog a + log sin y = log b+ log tan y. 

dog 4/4 (a —b") == $ [log (a+ 5) + log (a—B)). , 
log 4/ 2 +b?) =log a+-log sec u== log b+ log cosec u. 
log 4/(a-bb) = 4log a+ log sec x = Jloga + glog 2-+- log coszy. 
log 4/ (a — b) = 4 loga+t-log cos z= glog a-++ glog 2-+ log sin Sy. 


log (a. 5)” = “(log af-log cos ¢-log tan 45° + 34)). 


The first three of these formule will often be useful, when two sides of a night-an- 
zled triangle are given, to find the third. x? 


268 SOLUTION OF BIQUADRATIC EQUATIONS. 


, when 2pn--r 

x? tpe-tn=t a/ (p? (p?-F2n-F9)eb4/(n? +s) ; st potty 
when 2pn-+-r 

ore? tpar--n=t MP Figo (nt +s 3) } e resatipel 
And from these two quadratics,. the four roots of the given 
bignadratic may be determined*. 

Note. Whenever, by taking away the second term of a bi- 
quadratic, after the manner described i in cor. th. 8, the fourth 
term also vanishes, the roots may immediately be obtained by 
the solution of a quadratic only. 

_ A biquadratic may also be solved independently of cubics, 
in the following cases : 

1. When the difference between the coefficient of the third 
term, and the square of half that of the second term is equal to 
the coefficient of the fourth term, divided by half that of the se- 
cond. ‘hen if p be the coefficient of the second term, the equa- 
tion will be reduced toa quadratic by dividing it by x? +1pzf. 

2. When the last term is negative, and equal to the square of 
the coefficient of the fourth term divided by 4 times that of the 
third term, minus the square of that of second : then to complete 
the square, subtract the terms of the proposed biquadratic from 
(x? 4px), and add the remainder to both its sides}. 

3.’ When the coefficient of the fourth term divided by 
that of the second term, gives for a quotient the square root 
of the last term: then to complete the square, add the square 
of half the coefficient of the second term, to twice the square 











* This rule, for solving biquadratics, by concelving each to be the difference of 
two squares, is frequently ascribed to Dr. Waring; but its original inventor was 
Mr. Thomfs Simpson, formerly Professor of Mathematics i in the Royal Military 
Academy. 

f in this place the apo refers to a iqeatnes of the form 

at + 2an? 4 (a? - he -+- abe — c. : 
ef which the left kan side is divisible by x? ax, and the cit is 2 + ax 
-+ 25, so that the paietion. becomes 
; (2% =p ax). can + a +20) ao. 

Ox, we may complete the Hale in the proposed ecuation by adding 5? to each 

side, and we ie 


w* p2ae? (a? +25 +25)e 2 42aba- 5? mas 


aeies x -har-bb = = Si ee, c, Ed. 
t In this place ue author t gives the snes of the Pree 
4 f-2ax* + (22m 2) #7 =2mnxz—n? —0, 
Wend 
which gives ash =—n?, 
a? —m* )—(2a)” 
This biquedratic, by following a directions of the author, or more. simply by 
transposition, Recomet 
4 24x? a2 2? == m7 a7 +2mnztn?, - 
from which, by tate sige square root, we have the quadratic equation 
x wr ax = (ma +n). ee FOOt 


EULER’S RULE FOR BIQUADRATICS. 269° 


root of the last term, multiply the sum by 2x?, from the pro- 
duct take the third term, and add the remainder to both sides 
of the biquadratics.* bite ce eee 
4. The fourth term will be made to go out by the usual 
operation for taking away the second term, when the differ- 
ence between the cube of half the. coefficient of the second 
term and half the product of the coefficients of the second 
and third term, is equal to the coefficient of the fourth term. 
IV. Euler’s Rule for the Solution of Biquadratics. 
Let x4 --ax? —bx —c=0, be the given biquadratic equation 
wanting the second term. Take f= 1a, ¢ = ja? + 1c, and 
h = 31,62, or 4/h=10: with which values of f, g, h, form the 
cubic equation, 2? — fz2-+ gz-—h=0. Find the roots of this 
cubic equation, and let them be called p, 9,7. Then shall 
the four roots of the proposed biquadratic be these following : 
Viz. | 
When 15 is positive. . | When 10 is negative : 
Let VS PESITST yf = SPT V/I—V/"- 
2.2 of P—Vfq-a/"- BS SPH VI To" 
3. 2S — A/Prof T9-Vf?- B= —A/PE/ F/T 
4.0 —S/pr—SfGtJt | BE SPI" 
Note. 1. In any biquadratic equation having all its terms, 
if 2 of the square of the coefficient of the 2d term be greater 
than the product of the coefficients of the Ist and 3d terms, 
or 3 of the square of the coefficient of the 4th term be great- 
er than the product:of the coefficients of the’3d and _ fifth 
terms, or 4 of the square of the coefficient of the 3d term 
greater than the product of the coefficients of the 2d and 4th 
terms ; then all the roots of that equation will be real and un- 
equal; but if either of the said parts of those squares be less 
than either of those products, the equation will have imagi- 
nary roots. : 
2, In a biquadratic x4-+- ax3-++ ba? -+- cx + d=0, of which 
two roots are impossible, and d an affirmative quantity, then 
the two. possible roots will be both negative, or both affirma- 
tive, according as a? — 4ab-4-8c, is an afhrmative ora negative 
quantity, if the signs of the coefficients a, 6, c, d, are neither 
all affirmative, nor alternately — and +-. 





* The equation is «* -+2am? 4 pax? +-2aba + b? — 0, 


which is evidently «* -- 2aa? (a? 4. 2b)x? +-2abw-f.b7 — (a7 -2b—n) 2”, 
from which by taking the root we have the quadratic 


Sa oe ao? an b tb af a? 2h—p. : 

+ Various general rules for the solution of equations have been given by De- 
moivre, Bezout, Lagrange, &c.; but the most universal in their application are 
<ppreneee rules, of which avery simple and useful one is given in our first 
yoiume, ; - y y 





EXAMPLES, 


970 NUMERAL SOLUTIONS OF EQUATIONS. 


“EXAMPLES. 


Ex. 1, Find the roots of the equation x? + aes wi TONG 


~ 127) 12716 
by tables of sines and tangents. 


od 


7 1695 
Here p= od eee and the equation agrees with the 


88." 1695 ne 1695 


ist, form. Also tan a=— > V son76? and 7= tan a= / T5916" 


In logarithms thus : 
Log. 1695 == 3-2291697 


Arith. com. log. 12716 = 5°8956495 

sum + 10 = 19-1248192 

half sum = 9-4624096 

log 88 = = 1°9444827 

Arith. com. log. 7 = 9-1549020 
sum — 10 = logtana = 10°6617943 = log tan 77°42'31"3 ; 

log tanta = 9-9061115 = log tan 38°51’ 1B, : 

log ./ 9; as above = -9:5624096 


sum — 10 = log z= — ‘14685211 = log 2941176. 
This value of 2, viz. :2941176, is nearly equal to ne To find 


whether that is the exact root, take the arithmetical compli- 
ment of the last logarithm, viz. 0°5314379, and consider it as 
the logarithm of the denominator of a fraction whose nume- 


rator is unity ; thus is the fraction found to be 3.4ekactly, and — 


this is manifestly equal to “ee As to the other root of the 
mate 5 to 2 1608. 8) 389 | | 
Ex. 2. Find the roots of the cubic eee 
403 46 


sicmme EE od + fies 0, by a table of sines. 
2 403 6 
_ Herep = a q= Pi the second term is: negative, and 


4n3 > 2792: sothat the example falls under the irreducible case. 
Hesnd vingum cet 
Seen ee 147” 40a" 408403" 1612" 
Q/SA41 4/1328 
The three values of x therefore, are 
1612 
“= sin aA/———; 1303" 


C= gin 


SOLUTIONS OF EQUATIONS. O11 
| 1612 
ey ? _. j 
—e sin (60 ~4) V F305 
1612 
x =—sin (60° A) of ss55 a5" 
The logarithmic computation is subjoined. 
3 Log 1612 = 3°2073650 
Arith. “com. log 1323 = 6-8784402 
sum — 10..... = 0:0858052 
half sum = 0:0429026 const. log. 
Arith. com. const. log. == 9:9570974 
log 414. . . = 2°6170003 
Arith. com. log. 403 . = 7°3946950 
log sin 3a . = 9-9687927 = logsin 68°32 18"1, 
Log sin a = 9° 5891206 
. const. log = 0:0429026 ; 
1. sum — 10 = log x = — 1-6320232 =log4285714=log?. 
Log sin (60°—a) = 9:°7810061 LAG 
. const. log . . . . = 0:0429026 
2. sum — 10 = log « = — 1:8239087 =log’ BURP BRO Te ee: 
Log sin (60°-+a) = 9-9966060 
const. log... . = 0:0429026 


3. sum — 10=log —~x = 0- 0-0395086—logt: 095238=log2 2 
So that the three roots are 2 2, 2, and —23;. of which the first 


two are together equal to the third with its sign changed, as 
they ought to be. 


Ex. 3. Find the roots of the biguadratie x4— 25a? -1600— 
36==0, by Euler’s Rule. 


Here a=25, b=— 60, and c=36 ; therefore 
25 sate 7 ay 295 
fepeHzetI= zp adh= —. 
Consequently the ie equation wil be | 
25 too 225 
SD pce cee eo een 
z 3 z2 +4 — ; eye | 0. 


The three roots of which are 
9. 25 - 
2=2=P, and 2-44; and z= ang ; 


the square one of these are JS P= 3, f/9=2 OF 4, fr= 
Hence, as the value of 105 is negative, the four roots are 


Ri 


Ist.o = 244-§= 1, 
2d.e=> 3—4+45=>= 2, 
3d. 2 = —3+44+45= 3, 
4th.x = —3~4--4=—6.. 


Ex. 4 


279 - SOLUTIONS OF EQUATIONS. 


Ex. 4. Produce a quadratic equation whose roots shall be 
¢ and 4. . Ans. 2? ~3144+2=0. 
Ex. 5. Produce a cubic equation whose roots shall be, 2 
5, and —3. Ans. x3? —4¢2—~] 1x-+30=0. 

Ex. 6. Producea biquadratic which shall have for the roots 
1, 4,—5, and 6 respectively. | 
Ans. aot — 623 — 212? +1462— 120=0. 


Ex. 7. Find x, when «?-+-3477=22110. 
Ans. 2=55, re=—402. 


5f 
Ex. 8. Find the roots of the cua e3— 3 t= Peat 
65 
An =10, x= —-—. 
ns. x A 12 

A Si | 260.2 Be 

Ex. 9. Solve the equation x? — thd t= — Pag, 

Ans. 25, 2= a 


Ex. 10. Given a* ae 241132 — 481860, to find z. 
‘Anis. x==20 1=24093. 
Ez. 11. Find the roots of the equation x3 —32 —1=0 
Ans. the roots are sin 70°, — sin 50°, and — sin 10°, £6 a 
radius = 2; or the roots are twice the sines of those arcs as 
given in the tables. 


Ez. 12. Find the real root of Welk nid 25g 
Ans. 2,/3X sec. 54° 44 20". 


_ Ex. 13. Find the real root of 2523-+-+-'75xz — 46=0. 
Ans: 2 cot 74° 27’ 48”, 


Ee 14. Given t4 — 8a? — 12x? -+-84a - 63=0, to find x x by 
quadratics. Ans. e=2+ VTzWV1 TIT. 
Ex. 15. Given x4 +3622 —400x? — 31682 + 7744 = 0, to 


find x, by quadratics. Ans. x=11-+ 209. 
Ex. 36. Given x4 +2473 — 11442 —247-+-1=0 to find z. 
Ans, a= 4/197—14, c=2+ 5. 
Ex. 17. Find 2, when x! — 12x—5=0. 
Ans. © =1+,/2, c=—142/ ~1. 


Ex. 18. Find a, when x* — 1229-4 7x2 —724-+4-36—0. 
Ans. «=1, or 2, or 3, ily 


Ex. 19. Given xf = bart —-80@) a? = 68a ta aya ig 


to find z. 
Ans. 2 —a, n= bata y 37, aay 10—3a. 


ON 


{ 273 ] 


ON THE NATURE AND PROPERTIES OF CURVES, 
AND THE CONSTRUCTION OF EQUATIONS: 


SECTION I. 
Nature and Properties of Curves: 


Der. 1. A curve is a line whose several parts proceed in 
different directions, and are successively posited towards dif- 
ferent points in space, which also may be cut by one right line 
in two or more points. 

If all the points in the curve may be included i in one plane, 
the curve is called a plane curve; but if they cannot all be 
comprised in one plane, then is the curve one of double cur- 
vature. 

_ Since the word direction implies straight lines, and in strict- 
ness no part of a curve is a right line, some geometers prefer 
defining curves otherwise: thus, ina straight line, to be called 
the line of the abscissas, from a certain point let a line arbi- 

_ trarily taken be called the abscissa, and denoted (commonly) 
by x: at the several points corresponding to the different 
values of 2, let straight lines be contiaually drawn, making a 
certain angle with the line of the abscissas : these straight lines 
being regulated in length according to a certain law or equa- 

. tien, are called ordinates; and the line or figure in which 
their extremities are continually found is, in general, a curve 
line. This definition however is not free from objection ; 
for a right line may be denoted by an equation between its 
abscissas and ordinates, such as y=ax-+-b. 

Curves are distinguished into algebraical or geometrical, 
and transcendental or mechanical. ¢ 


Def. 2. Algebraical or geometrical curves, are those in 
which the relations of the abscissas to the ordinates can be 
denoted by a common algebraical expression; such, for ex- 
ample, as the equations to the conic sections, given in page 
632, &c. of vol. 2. 

Def. 3. Transcendental or mechanical curves, are such as 
- cannot be so defined or expressed by a pure algebraical equae 
tion ; or when they are expressed by an equation, having one 

Vou. IL. 36 of 


i xt 


274 NATURE AND PROPERTIES OF CURVES. 


of its terms a variable quantity, or a curve line. Thus, y = 
loga,y = A. sin x, y= . COS x, y=a"*, are Equations to tran- — 
scendental curves ; and the latter i in panels is an equation 
to an exponential curve. 

Def. 4. Curves that turn round a fixed point or centre, 
gradually receding from it, are called spiral or radial curves, 

Def. 5. Family or tribe of curves, is an assemblage of 
several curves of different kinds, all defined by the same 

equation of an indeterminate degree ; ; but differently. accord- 

ing to the diversity of. their kind. For example, suppose an 
equation of an indeterminate degree, a”—!z=y™: if m=2, 
then will ag=y? ; if m=3, then will a22=y? ; ifm=4,then ~ 
is a°a=y*, &c.: all which curves are said to ie of the same 

family or tribe. 

Def. 6. The avis of ‘a figure is a pre: line passing sbiatiob 
the centre of a curve, when it has one: if it bisects the or- 
dinates, it is called a diameter. 

Def.7. An asyinptote is a-right line pasa continually ap- 
proaches towards a curve, but never can touch it, unless the 
curve could be extended to an mfinite distance... 

Def. 8. An abscissa and an ordinate, whether right.« or obs 
lique, are, when spoken of bogethes, frequently termed co- 
‘ordinates. 

Art, I. The most convenient mode of classing algebraical 
curves is according to the orders or dimensions of the equa- 
tions which express the relation between the co- -ordinates, 
For then the equation for the same curve, remaining always 
of the same order so long as each of the assumed systems of 
co-ordinates is supposed to retain constantly the same inclina- 
tion of ordinate to abscissa, while referredo different points 
of the curve, however the axis and the origin of the abscissas,.. 
or even the inclination of the co-ordinatesin different systems, 
may vary; the same curve will never be ranked under dif- 
ferent orders, according to this method. If therefore we take, 
for a distinctive character, the number of dimensions which 
the co-ordinates, whether rectangular or oblique, form in the 
equation, we shall not disturb the order of the classes, by chang- 
ing the axis and the origin of the abscissas, or by varying the 
Bichnation of the co-ordinates. 

2. As algebraists call orders of different kinds of equations, 
those which constitute the greater or less number cf dinen- 
sions, they distinguish by the same name the different kinds 
of resulting lines. Consequently the general equation of the 
first-order being 0 = «+ 6x-+ vy; we may refer to the 
first order all the lines which, by teking x and y for the co- 
ordinates, whether rectangular or oblique, give rise to this 

equation. 


NATURE AND PROPERTIES OF CURVES. 975 


equation, But this equation comprises the Tight line alone, 
which is the most simple of all lines ; and since, for this rea- 
son, the name of‘curve does not properly apply to the first 
order, we do not usually distinguish the. different orders by 
the name of curve lines, but simply by the generic term of 
lines : hence the first order of lines does not compr ehend any 
curves, but solely the right line. — 

As for the rest, it is indifferent whether the co-ordinates 
are perpendicular or not; for.if the ordinates make with the 
axis. an angle 9 whose aiae is ~# and cosine », we can refer the 


genta to ce of the rectang gular co-ordinates, by making — 


y= sy and x=" +t; which will give for an equation be- 
fe : } mes oe: ! 
tween the perpendiculars ¢ and wu. 


O=atar+ (247). 


Thus it follows evidently, that the signification of the equa- 
tion is not limited by supposing the ordinates to be rightly ap- 
plied: and it will be the same with equations of superior or- 
ders, which will not be less general though the co-ordinates 
are perpendicular. Hence, since the determination of the 


inclination of the ordinates applied’ to the axis, takes nothing 


from the generality of a general equation of any order what- 
ever, we put no restriction on its signification by supposing 
the co-ordinates rectangular ; and the equation will be of the 
same order whether the co-ordinates'be rectangular or oblique. 

3. All the lines of the second order will be comprised in 
the general equation. 

O==a+prtyy+ dx? hoc Ege 7 
that is to say, we may class among lines of the seroutl order 
all the curve lines which this equation expresses, x and y de- 
noting the rectangular co-ordinates. ‘These curve lines are 
therefore the most simple of all, since there are no curves in 
the first order of lines ; it is for this reason that some writers 
call them curves of the first order. But the curves included 
in this equation are better known under the name of conre 
sections, because they all result from sections of the cone. 
The different kinds of these lines are the ellipse, the circle, 
or ellipse with equal axes; the parabola, and the hyperbola ; 
the properties of all which may be deduced with facility from 


the preceding general equation. Or this equation may be - 


transformed into the subjoined one : 


" pty as A ee 


=0; 








and 


* 


NATURE AND PROPERTIES OF CURVES. 


and this again may be reduced to the still more simple ~ 
yr=fr?-gee+h. 

_ Here, whan the first term fx? is affirmative, the curve ex- 

pressed by the equation is a hyperbola; when fx? is negative 

the curve is an ellipse; when that term is absent, the curve 

is a parabola. When x is taken upon a diameter, the equa-’ 

tions reduce to those already given in sec. 4 ch. i. 

The mode of effecting these transformations is omitted for 
the sake of brevity. -fhis section contains a summary, not: 
an investigation of properties: the latter would require many 
volumes, instead of a section. 


4.. Under lines of the third order, or curves of the second, 
are classed all those which may be expressed by the equation 
O= a + Bx byy+da2 + exy ly? + nx + bx2y+iay? + xy? 
And in like manner we regard as lines of the fourth order, 
those curves which are furnished by the general equation 
O=a-+ Bx t+ yy dx? + exy + Cy? + 2x3 + bx2y + xy? +- 

3 ey8-b arte maty 4+ vty? + Exy3 oy! ; 
taking always x and y for gectaneaay co-ordinates. In the 
most general equation of the third order, there are 10 con- 
stant quantities, and in that of the fourth order 15, which 
may be determined at pleasure; whence it results that the 
kinds of lines of the third order, and,.much more, those of 
the fourth order, are considerably more numerous than those 
of the second. ‘ 


5. It will now be easy to conceive, from what has gatie be- 
fore, what are the curve lines that appertain to the fifth, sixth, 
seventh, or any higher order; but as it is necessary to add to 
the general equation of the fourth order, the terms 

xe, c4y, xy?, x?y%, ry*, Y?, : 
with their respective constant co-efficients, to have the gene- 
ral equation comprising all the lines of the fifth order, this 
latter will be composed ef 21 terms: and the general equa- 
tion comprehending all the lines of the sixth order, will have 
28 terms ; and so on, conformably to the law of the triangu- 
lar numbers. ‘Thus ‘the most general equation for lines of the 


(tt vente Fecnie ) 


order n, will contain terms, and as many con- 


stant letters, which may ke aoe, at pleasure. 

6. Since the order of the proposed equation between the 
¢0- -ordinates, makes known that of the curve line ; whenever 
we have given an algebraic equation between the co- -ordinates 
x and y, or ¢ and uw, we know at once to what order itis ne- 
geasary to refer the curve represented by that.equation. If 

the equation be irrational, it must be freed frem radicals, and 
if 


NATURE AND PROPERTIES OF CURVES. 277° 


if there be fractions, they must. be made to disappear ; this 
done, the greatest number of dimensions formed by the varia- 
ble quantities « and_y, will indicate the order to which the 
line belongs. Thus the curve which is denoted by this equa- 
tion y? —az=0, will be of the second order of lines, or of the 
first order of curves; while the curve represented by the 
equation y?=2,/(a? —2?), will be of the third order (that is, 
the fourth order of lines), because the equation is of the fourth 
order when freed from rai ; and the line which is indi- 


aoe will be of the third order, 


or of the second order of curves, because the equation when 
the fraction is made to disappear, becomes a2 y+-22 y=a3 —az?, 
where the term x?y contains three dimensions. 

7. It is possible that one and the same equation may give 
different curves, according as the applicates or ordinates fall 
upon the axis perpendicularly or under a given obliquity. 
For instance, this equation, y2 =ax—x?, gives a circle, when 
the co-ordinates are supposed perpendicular ; but when the 
co-ordinates are oblique, the curve represented by the same 
equation will be an ellipse. Yet all these different curves ap- 
pertain to the same order, because the transformation of rect- 
angular into oblique co-ordinates, and the contrary, does not 
affect the order of the curve, or of its equation. »Hence, 
though the magnitude of the angles which the ordinates form 
with the axis, neither augments nor diminishes the generality 
of the equation, which expresses the lines of each order ; yet, 
a particular equation being given, the curve which it expresses 
can only be determined when the angle between the co-ordi- 
nates is determined also. 

__ 8. That a curve line may relate properly to the order in- 

dicated by the equation, it is requisite that this equation be 

not decomposable into rational factors ; for if it could be com- 

posed of two or of more such factors, ‘it would then compre- 

hend as many equations, each of which would generate a 

particular line, and the reunion of these lines would be all 

that the equation proposed could represent. Those equations 

then, which may be decomposed into such factors, do not 

comprise one continued curve, but several at once, each of 
which may be expressed by a particular equation ; and such - 
combinations of separate curves are denoted by the term com- 

plex curves. 

Thus, the equation y2= =ay--xy— ax, which seems to apper- 
tain to a line of the second order, if it be reduced to zere by 
making y? —ay— xy--ax = 0, will be composed of the factors 
(y—x) (y—a)=0; it therefore comprises the two equations. 

Y~x 


cated by the equation y= 





278 NATURE AND PROPERTIES OF CURVES. 


y—2=0, and y~a==0, both of which belong to the right line : 
the first forms with the axis at the origin of the abscissas an 
- angle equal to half a right angle ; and the second is parallel 
to the axis, and drawn at a distance =a, ‘These two lines, 
considered together, are comprised in the proposed equation 
y?==ay+zy—ar. In like manner we may regard as complex 
this equation y4 — acy? —a?x? ~ay>+ ax? y+-a02z2y=0; for its 
factors bemg (y—x) (y—a) (y2 —ax) = 0, instead of denot- 
ing oue continued line of the fourth order, it comprises three 
distinct os, viz. two rig bt lines, and one curve denotedby the 
equa. y? ~ az=v. 

9. i e@ may thitreliie form at pleasure any complex hinds 
whatever, which shail contain 2 or more right lines or curves. 
For, if the nature of each line is expressed by an ee re- 
ferred to the saine axis, and to the same | wy 
origin of the abscissas, and after having A 
reduced each equation to zero, we mul- 
_ tiply them one by another, there will By 
resulta complex equation which at once 
comprizes all the lines assumed. For 
example, if from the centre c, with a: 5 +7 (gan 
radius ca == a, acircle be described; and further, if a right 
line tw be drawn through the centre c ; then we may, for any 
assumed axis, find an equation which will at once include the 
circle and the right line, as thoagh these two lines formed on- 
ly one. 

Lannie there be taken for an axis the diameter AB, that 
forms with the right line Ly an angle equal to half a right 
angle ; having placed ‘the* origin of the abscissas in a make 
the abscissa ap==2, and the applicate or ordinate pm= y; 
we shall have for the right line, em=cr =a—.2; and since 
the point m of the right line falls on the-side of those ordi- 
nates which are reckoned negative, we have y=—a-+<«, 
or y-x-+a=0: but, for the circle, we have PM? = ap . pp, 
and sp = 2a — 2, which gives y2 = 2ax — #2, or y? + 1? ~ 
2ax=0.. Multiplying these two equations together we obtain 
the complex equation of the third order, 

ye i— yrx + yx? — 13 + ay? — Lary + 3axr2 — 2a°x = 0, 
which represents, at once, the circle and the rightline. Hence, 
we shall find that to the abscissa ap=z, corresponds three 
ordinates, namely, two for the circle, and one for the right 
line. Let, for example, s—=1a, the equation will pene, 
y?+ lay? —3a2y —2a3 = 0; whence we tie eae ytia= 

and by dividing by this root. we obtain y7— 3a? = 0, the pi 
roots of which being taken and ranked with ie former, ah 
the three ia values of y; 







tye 


- NATURE: i PROPERTIES OF CURVES. 279 


Ly= — Be 49 
AL yer jay 3. 
yo it Daas 1a,/3.° : ia 
We see Bie that the whole is represented vein one wna! 
tion, as if the circle together with the right line formed aaly 
one coritinued curve. 

10. This difference between simple and onneex curves 
heiod once established, it is manifest. that the lines of the se- 
cond order are either continued curves, or compiex lines 
formed of two right lines; for if the general equation have 
rational factors, they must be of the first order, and conse- 
quently will denote right lines. Lines of the third order will 
be either simple, or complex, formed either of a right line - 
and a line of the een order, or of three right lines. In 
like manner, lines of the fourth order will be continued and 
simple, or complex, comprising a right line and a hne of the 
third order, or two lines of the second order, or lastly, four 
right lines. . Complex lines of the fifth and superior orders 
will be susceptible of an analogous combination, and of a si- 
milar enumeration. Hence it follows, that any order what- 
ever of lines may comprise, at once, all. the lines of inferior 
order, that is to say, that they may contain a complex line of 
any inferior orders with one or more right lines, or with lines 
ef the second, third, &c. orders: so that if we sum the nam- 
bers of each order, appertaining to the simple lines, there 
will result the number indicating the. order of the complex 
line. | 
- Def. 9. That is daiher an hyperbolic leg, or branch of a 
curve, which approaches constantly to some asymptote ; and 
that a parabolic one which has no asymptote. | 

Arr. 11. All the legs of curves of the second and higher 
kinds, as well as of the first, infrnitely drawn out, will be of 
either the hyperbolic or the parabolic kind: and these legs 
are best known from the tangents. For if the point.of con- 
tact be at an infinite distance, the tangent of a hyperbolic leg 
will coincide with the asymptote, and the tangent of a para- 
bolic leg will recede tn infinitum, will vanish and be no where 
found. Therefore the asymptote of any leg is: found by seek- 
ing the tangent to that leg at a point ‘infinitely distant: and 
the course, or way of an infinite leg, is found by seeking the 
position of any right line which is parallel to the tangent 
where the point of contact goes off in infinitum: for this right 
line is directed the same way with the infinite leg. 


- Sir Isaac N ewton’s Reductioa of all Lines of the Third Or- 


ny to four Cases oF Equations ; ; a the i ah of those 
ines. 5; 


CASE i 


280 LINES OF THE THIRD ORDER. 


CASE I. 


12. All the lines of the first, third, fifth, and seventh order, 
er of any odd order, have at least two legs or sides proceed- 
ing on ad infinitum, and towards contrary parts. And all lines 
of the third order have two such legs or branches running out 
contrary ways, and towards which no other of their infinite 
legs (except in the Cartesian parabola) tend. If the legs are 
of the ie hypervars kind, let cas be their asymptote ; and to it 





let the PUNT S cee be drawn, terminated (if possible) at both 
ends at the curve. - Let this parallel be bisected in x, and 
then will the locus of that point x be the conical or common 
hyperbola xe, one of whose asymptotes is as. Let its other. 
asymptote be as. ‘Then the equation by which the relation 
between the ordinate sc-=y, and the abscissa aB==a, is deter- 
mined, will always be of this form: viz. 
xy? +ey=ax3--bx2-+ce+d .. (I.) 
Here the coefficients e, a,b,c, d, denote given quantities, 
affected with their signs +- and —, of which terms any one 
may be wanting, provided the figure through their defect does 
not become transformed into a conie section. The conical 
hyperbola xq may coincide with its asymptotes, that is, the 
point x may come to be in the line 4B; and then the term -+ 
ey will be wanting. , 
CASE IE. « 

13. But if the right -line cBe cannot be terminated both 
ways at the curve, but will come to it only in one point; then 
draw any line in a given position which shall cut the asymp- 
tote asin a; as also any other right line, as sc, parallel to 

the 


LINES OF THE THIRD ORDER. 281 


the asymptote, and meeting the curve in the point c; then 
the equation, by which the relation between the ordinate sc 
and the abscissa aB is determined, will 1 ae, assume this 
form: viz. ry=ax?+be?-+-cx+d . » (EL) 


CASE IIL. 


14. If the opposite legs be of the parabolic kind, draw the 
right line csc, terminated at both ends (if possible) at the 
curve, and ranning according to the course of the legs ; which 
line bisect in 8: then shall the locus of s be a right line. Let 
that right line be as, terminated at any given point, as a: then 
the equation, by which the relation between the ordinate sc 
and the abscissa az is determined, will brian be of this form: 

y2=ar3-+-ba2-+cat+d . . (IIT) 


CASE IV. 


15. If the right line cbc meet the curve only in one point, 
and therefore cannot be terminated at the curve at both ends ; 
let the point where it comes to the curve be c, and let that 

right line at the point 8, fall on any other right line given in 
position, as as, and terminated at any given point, as a. Then 
will the equation expressing the relation between sc and ap, 
assume this form : 

y=an>-ba?-+crtd.... (IV.) 3 

16. In the first case, or that of equation. 1, if the term ax? 
be affirmative, the figure will be a triple hyperbola with six 
hyperbolic legs, which will run on infinitely by the three 
asymptotes, of which none are parallel, two legs towards each 
asymptote, and towards contrary parts ; and these asymptotes, 
if the term 6x? be not wanting in the equation, will mutually 
intersect each other in 3 points, forming thereby the triangle 

_pdéd, But if the term bx? be wanting, they will all converge 
to the same point. This kind of hyperbola is called redun- 
dant, because it exceeds the conic hyperbola in ine number 
of its hyperbolic legs. 

In every redundant hyperbola, if neither the teri ey be 
wanting, nor b? — 4ac = ae,/a, the curve will have no diame- 
ter; butif either of those occur separately, it will have only 
one diameter ; and three, if they both happen. Such diame- 
ter will always pass through the intersection of two of the 
asymptotes, and bisect all right lines which are terminated 
each way by those asymptotes, and which are parallel to the 
third asymptote. 

17. If the redundant hyperbola have no diameter, let the 
four roots or values of x in the equation ax4-+-bx?+cx? +dx 
+ 12 =O, be sought; and suppose them to be ar, aw. Am, 

Vou. IL. 37 and 


a8 LINES OF THE THIRD ORDER. 


and ap (see the preceding figure). Let the ordinates rv, wr, 
#1, pt, be erected ; they shall touch the curve in the points 
 T, 4%, 7, t, and. by that contact shall give the limits of the curve, 
by which its species will be discovered. 

Thus, if all the roots ap, aw, ax, ap, be real, and have the 
same sign, and are unequal, the curve will consist of three 
hyperbolas and an oval: viz. an inscribed hyperbola as ec 3 a 


- ewrcumseribed hyperbola, as Tdc; and ambigeneal hyperbola, 


_ (i.e. lying within one asymptote and beyond another ) as pt ; 

and an oval +7. This is reckoned the first species. Other 
relations of the roots of the equation, give 8 more different 
species of redundant hyperbolas without diameters ;. 12 each 
with but one diameter ; 2 each with three diameters; and 9 
each with three asymptotes converging to a common point. 
Some of these have ovals, some points of decussation, and in 
some the ovals degenerate into nodes or knots. 

18. When the term ax? in equa. 1, is negative, the figure 
expressed by that equation, will be a deficient or defective hy- 
perbola ; that is, it will have fewer legs than the complete 
conic hyperbola. Such is the marginal 
figure, representing Newton’s 33d spe- 
cies ; which is constituted of an angui- 
neal or serpentine hyperbola, (both legs 
approaching a common, asymptote by 
means of a contrary flexure, and a con-. 
jugate oval. There are 6 species of de- 
fective hyperbolas, each having but one. 
asymptote, and only two hyperbolic legs, 
running out contrary ways, ad infini- 
tum; the asymptote being the first and principal ordinate, 
When the term ey is not absent, the figure will have no dia- 
meter ; when it is absent, the figure will have one diameter. 
Of. this latter class there are 7 different species, one of which, 
namely Newton’s 40th species, is exhibited in the margin. 

19. If, in equation 1, the term ax? . ace 
be wanting, but bx? not, the figure ex- as 3 | 
pressed by. the equation remaining, 
will be a parabolic hyperbola, having : 
two hyperbolic legs to one asymptote, | _ CoN ols \ 
and two parabolic legs converging one © © ST | A] 
and the same way. When the term ey 
is not wanting, the figure will have no : 
diameter ; if that term be wanting, the | 
figure will have one diameter. There 
are 7 species appertaiene to the former case ; and 4 to the 
latter. 








20. When 


LINES OF THE THIRD ORDER. 283 


20, When, in equa. 1, the terms ax?, ba?, are wanting, or 
when that equation becomes xy?-+-ey=ca--d, it expresses 
a figure consisting of three hyperbolas opposite to one an-. 
other, one lying between the parallel asymptotes, and the 
other two without: each of these curves having three asymp- 
totes, one of which is the first and 
principal ordinate, the other two pa- 
rallel to the abscissa, and equally 
distant from it; as in the annexed 
figure of Newton’s 60th species. 
Otherwise the said equation ex- 
presses two opposite circumscribed — 
hyperbolas, and an anguineal hyper- 
bola between the asymptotes. Under 
this class there are 4 species, called 
by Newton Hyperbolisme of an hyperbola. By hyperbolisme 
of a figure he means to signify when the ordinate comes out, 
by dividing the rectangle under the ordinate of. a given conic 
section and a given right line, by the common abscissa. 

21. When the term cz? is negative, the figure expressed 
by the equation xy?--ey=—«czx?-+d, is either a serpentine 
hyperbola, having only one asymptote, being the principal 
ordinate ; or else it is aconchoidal figure. Under this class 
there are 3 species, called Hyperbolisme of an epee. 

22. When the term cz? is absent, the equa. xy? + ey =d, 
expresses two hyperbolas, lying, not in the opposite angles of 
the asymptotes (as in the conic hyperbola), but in the adja- 
cent angles. Here there are/only 2 species, one consisting of 
an inscribed and an ambigeneal hyperbola, the other of two 
inscribed hyperbolas. These two species are called the Hy- 
perbolisme of a parabola. 

23. In the second case of equations, or that of equation 11, 
there is but one figure ; which has four infinite legs. Of 
these, two are hyperbolic about one asymptote, tending to- 
wards contrary parts, and two converging parabolic legs, 
making with the former nearly the figure of a érident, the fa- 
miliar name given to this species. This is the Cartesian para- 
bola, by which equations of 6 dimensions are sometimes con- 
structed : it is the 66th species of N ewton’ 3 enumeration. 





24. The third case of equations, or 
equa 11, expresses a figure having two ae 
rabolic legs running out contrary ways : 
of these there are 5 different Poe A ) 
called diverging or bell- -form parabolas ; de 
of which 2 have ovals, 1 is nodate, ‘1 “i 
punctate, and 1 cuspidate. The figure shows Newton’s 67th 

species ; 





— B84 LINES OF THE FOURTH ORDER. 


species ; in which the oval must always be so small that ‘ne 
right line which cuts it twice can cut the — curve ot 
more than once. 


25. In the case to which equa. a 
refers, there is but one species. _It ex- 
presses the cubical parabola with con- 
trary legs. This curve may easily be 
described mechanically by means of a 
square and an equilateral hyperbola. Its 
most simple property is, that rm (paral- — 
lel to ag) always varies as avn? —Qr?. 

26. Thus according to Newton there are 72 species of lines 
of the third order. But Mr. Stirling discovered four more 
species of redundant hyperbolas ; and Mr. Stone two more 
species of deficient hyperbolas, expressed by the equation 
yx? =bx? +-cz-+d ; 1. e. inthe case when bx? -+-crz-+-d=0, has 
two unequal negative roets, and in that where the equation 
has two equal negative roots. So that there are at least '78 
different species of lines of the third order. Indeed Euler, 
who classes all the varieties of lines of the third order under 
16 general species affirms that they comprehend more than 80 
varieties ; of which the preceding enumeration necessarily 
comprises nearly the whole. 

27. Lines of the fourth erder are divided by Euler into 
146 classes ; and these comprise more than 5000 varieties : 
they all flow from the different relations of the quantities in 
the 10 general equations subjoined. 

ley? bfx2y? + gry3 thar y+iy? bhay-+ly 

2.44 fry? -+gxtybhay? piny-tky .. | 
02g? fy? gat ythy? phy oy cw & et eee 
» 09 ytat fy pe phay bey. a ef ex2 data. 
RE belts ool (0 bli cel co ps el Oe CN gv ! 
ys -bfay? +gry hy vos 
ys bendy fay? Lgay? hy? tinythy ) mo 
.a3y ery? -Lfr2y Tee +hayiy (ax? -+-b22 + 
‘ ney ey? fry? berg isphy we eG ended . 
10. #y -rey® -hfy? +gey thy - . «} 

98, ldies of-the fifth and higher orders, of necessity be- 
come still more numerous ; and present too many varieties to 
admit of any classification, at least in moderate compass. 
Instead, therefore, of dwelling upon these ; we shall give a 
concise sketch of the most curious and important properties 
of curve lines in general, as they have been deduced from a 
contemplation of the nature and mutual relation of the roots 
of the equations representing those curves. Thus a curve 

being 





nee O Or 9 


GENERAL PROPERTIES OF CURVES. 285 


being called of n dimensions, or a line of the nth order when 
its representative equation rises to m dimensions ; then since 
for every different value of x there are 7 values of y, it will 
commonly happen that the ordinate will cut the curve in n or 
in n—2,n—4, &c. points, according as the equation has n, or 
n—2, n—-4, &c. possible roots. It is not however to be in- 
ferred that a right line will cut a curve of n dimensions, in 2, 
n—2, n—4, &c. points only ; for if this were the case, a line 
of the 2d order, a conic section for instance, could only be cut 
by a right line in two points ;—but this is manifestly incorrect, 
for though a conic parabola will be cut in two points by a right 
line oblique to the axis, yet a right line parallel to the axis 
can only cut the curve in one point. 

29. It is true in general, that lines of the m order cannot 
be cut by a right line in more than m points ; but it does not 
hence follow, that any right line whatever will cut in n points 
every line of that order; it may happen that the number of 
intersections is n—1, n—2,n—3, &c. ton—n. The number 
of intersections that any right line whatever makes with a 
given curve line cannot therefore determine the order to 
which acurve line appertains. For,as Euler remarks, if the 
number of intersections be =n, it does not follow that the 
curve belongs to the order, but it may be referred to some 
superior order ; indeed it may happen that the curve is not 
algebraic, but transcendental. This case excepted, however, 
Euler contends that we may always affirm positively that a 
curve line which is cut by a right line in » points, cannot be- 
long to an order of lines inferior tox. Thus, when a right 
line cuts a curve in 4 points, it is certain that the curve does 
not belong to either the second or third order of lines ; but 
whether it be referred to the fourth, or a superior order, or 
whether it be transcendental, is not to be decided but by ana- 

lysis. : 

a4 30. Dr. Waring has carried this enquiry a step further than 
Euler, and has demonstrated that there are curves of any 
number of odd orders, that. cut a right line in 2,4, 6, &c. 
points only ; and of any number of even orders that cut a 
right line in 3, 5, 7, &c. points only ; whence this author like- 
wise infers, that the order of the curve cannot be announced 
from the number of points in which it cuts a right line. See 
his Proprietates Algebraicarum Curvarum. 

31. Every geometrical curve being continued, either re- 
turns into itself, or goes on to an infinite distance. . And if any 
plane curve has two infinite branches or legs, they join one 
another either at a finite, or.at an infinite distance. 

32. In any curve, if tangents be drawn to all points of the 

| curve ; 


286 GENERAL PROPERTIES OF CURVES. 


curve; and if they always cut the abscissa at-a finite distance 
from its origin ; that curve has an asymptote, otherwise, not. 

33. A line of any order may have as’ Siasbe asymptotes as 
it has dimensions, and no more. 

34. An asymptote may intersect the curve in so many powsd 
abating two, as the equation of the curve. has dimensions: 

Thus, in a conic. section, which is the second order.of lines, 
the asymptote does not cut the curve at alk; in the third or- 
der it can only cut it in one ‘point ; in the founth order in two 
points ; and so on. 

35. If a curve have as many asymptotes, as it has dimen- 

sions, and a right line be drawn to cut them all, the parts of 
that measured from the asymptotes to the curve, will together 
be equal to the parts measured in the same ateretinn, from 
the curve to the asymptotes. 
'. 36. Ifa curve of n dimensions have n asymptotes, then the 
content of the n abscissas will be to the content of the nm ors 
dinates, in the same ratio in the ‘curve and asymptotes ; the 
sum of their n subnormals, to ordinates perpendicular to their 
abscissas, will be equal to the curve and the asymptotes ; ; and 
they will have the same central and diametral curves. 

37. If two curves of n and m dimensions have a common 
asymptote ; or the terms of the equations to the curves of the 
greatest dimensions have a common divisor ; then the curves 
cannot intersect each other in n Xm points, possible or impos- 
sible. If the two curves have a common general centre, and 
intersect each other in x X m points, then the sum of the af- 
firmative abscissas, &c. to those points, will be equal to the 
sum of the negative ; and the,sum of the » subnormals toa 
curve which has a general centre, will be Proportional to the 
distance from that centre. 

38. Lines of the. third, fifth, seventh, &c. order, or any 
odd. number, have, as before remarked, at least two infinite 
legs or branches, running contrary ways ; while in lines of 
the second, fourth, sixth, or.any even number of dimensions, 
the figure may return into itself, and be paren within cer- 
tain limits. 

39. If the right lines AP, PM,” forming 2 a given angle, arm 
cut a geometrical line of any order in as many points as it 

has dimensions, the product of the segments of the first ter- 

minated by p and the curve, will always be to the product: of 
the segments of the latter, terminated by the same point, and 
the curve, in an invariable ratio. 

40. With respect to double, triple, quadruple, and other 
multiple points, or the points of intersection of 2, 3, 4,-0r 
more dbeanches of acurve, their nature and number may be 

1 estimated 


GENERAL PROPERTIES OF CURVES, 287 


estimated by means of the following principles. 1..A curve of 
the n.order is determinate when it is subjected to pass through 


ink) ler (n-+2) 


the number —1 points. 2. A curve of the n 


order cannot intersect acurve of the m order i in more than 
mn points. = = 

Hence it falléwe that a curve of the gaeond order, for ex- 
ample, can always pass through 5 given points (not in the 
same right line), and cannot meet a curve of the m order in 
more than ma points ; and it is impossible that a curve of the 
m order should have 5 points whose degrees of multiplicity 
make together more than 2m points. Thus,.a line of the 
fourth order cannot have four double points ; because the line 
of the second order which would pass through these four 
double ‘points, and through a fifth simple point of the curve 
ef the: fourth dimension, would meet 9 times; which is im- 
poss ble, since: sere can ates be an intersection 2 x 4 or 8 
times. 

For the same reason, a curve Nine of the fifth cannbt) with 
ene triple point, have more. than three double points : and in 
- a similar manner we may reason for curves of higher orders. 

Again, for the known proposition, that we can always make 
aline of the third order pass through nine points, and that a 
curve of that order cannot meet a curve of the m order in more 
that 3m points, we may conclude that a curve of the m order 
cannot have nine points, the degrees of-multiplicity of which. 
make together a number greater than 3m. Thus, a line of the 
fifth order cannot have more than 6 double points ; a line of 
the 6th order, which cannot have more than one quadruple 
point, cannot have with that quadruple point more than 6 dou-_ 
ble points; nor with: two triple points more than 5 double 
points ; nor even with one triple point more than 7 double - 
points. Analogous conclusions obtain with respect to a line 

of the fourth order, which we may cause to pass through !4 
points and which can only meet a curve of the m order i ink 
ped eee and so on. 

The properties of curves of a superior abdery agree, 
ine certain modifications, with those of all inferior orders. 
For though some: line or lines become evanescent, and others 
become infinité, some coincide, others become equal ; some 
points coincide, and others are removed to an infinite distance ; 
yet. under these circumstances the general properties still 
hold good with regard to the remaining quantities ; so that 
whatever is demonstrated generally of any order, holds true 
. in the: inferior orders: and, on the contrary, there is hardly 
any property of the. inferior orders, but there is some similar 
to it, in the superior ones. For, 


288 GENERAL PROPERTIES OF CURVES. 


_ For, as in the conic sections, if two parallel lines are drawn 
fo terminate at the section, the right line that bisects these 
will bisect all other lines parallel to them; and is therefore 
called a diameter of the figure, and: the bisected lines ordi- 
nates, and the intersections of the diameter with the curve 
verticis; the common intersection of all the diameters. the 
centre ; and that diameter which is perpendicular to the or- 
dinates, the vertex. So likewise in higher curves, if two 
parallel lines be drawn, each to cut the curve in the number 
of points that indicate the order of the curve ; the right line 
that cuts these parallels so, that the sum of the parts on one 
side of the line, estimated to the curve, is equal to the sum 
of the parts on the other side, it will cut ‘in the same manner 
all. other lines parallel to them that meet the curve in the 
same number of points ; in this case also the divided lines are 
called ordinates, the line so dividing them a diameter, the in- 
tersection of the diameter and the curve vertices; the common 
intersection of two or more diameters the centre; the diame- 
ter perpendicular to the ordinates, if there be any such, the 
axis; and when all the. diameters concur in one point, that 4 1s 
the general centre. 

Again, the conic tise being a line of the cate or- 
der, has two asymptotes; so likewise, that of the third order 
may have three; that of the fourth, four; and so on; and 
they can have no more. And as the parts ‘of any right line 
between the hyperbola and its.asymptotes are equal ; so like- 
wise in the third order of lines, if any line be drawn cutting 
the curve and its asymptotes in three points ; the sum of two 
parts of it falling the same way from the asymptotes to the 
curve, will be equal to the part falling the contrary way from 
the third asymptote to the curve ; and so of higher curves. 

Also, in the conic sections which are not parabolic: as the 
square of the ordinate, or the rectangle of the parts of it on 
each side of the diameter, is to the rectangle of the parts of 
the diameter, terminating at the vertices, in a constant ratio, 
viz. that of the latus rectum, to the transverse diameter. So 
in non-parabolic curves of the next superior order, the solid 
under the three ordinates, is to the solid under the three ab- 
scissas, or the distances to the three vertices ; in a certain giv- 
en ratio. In which ratio if there be taken three lines propor- 
tional to the three diameters, each to each ; then each of these 
three lines may be called a latus rectum, and each of the cor- 
responding diameters a transverse diameter. And, in the 
common, or Apollonian parabola, which has but one vertex 
for one diameter, the rectangle of the ordinates is equal to 
the rectangle of the abscissa and latus rectum ; so, in those 

curves 


NATURE AND EQUATIONS OF CURVES. 289 


curves of the second kind, or lines of the third kind which 
have only two vertices to the same diameter, the solid under 
the three ordinates, is equal to the solid under the two ab- 
scissas, and a given line, which may be reckoned the latus 
rectum. 

Lastly, since:in the conic sections where two parallel lines 
terminating at the curve both ways, are cut by two other pa- 
rallels likewise terminated by the cure; we have the rect- 
angle of the parts of one of the first, to the rectangle of the 
parts of one of the second lines, as the rectangle of the parts 
of the second of the former, to the rectangle of the parts of 
the second of the latter. pair passing also through the com- 
‘mon point of their division. So, when four such lines are 
drawn ina curve of the second kind, and each meeting it in 
three pomts ; the solid under the parts of the first line, will 
be to. that under the parts of the third, as the selid under the 
parts of the second, to that under the parts of the fourth. 
And the analogy between curves of different orders may be 
carried much further: but as enough is given for the objects 
of this work; we shall now present a few of the most useful 
problems. 


PROBLEM f. 


Knowing the Characteristic Property, or the Manner of De- 
scription of a Curve, to find its Equation. 


This in most cases will be a matter of great simplicity ; be- 
cause the manner of description suggests the relation between 
the ordinates and their corresponding abscissas ; and this re- 
lation when expressed algebraically, is no other than the equa- 
tion to the curve. Examples of this problem have already oc- 
curred in sec. 4 of vol. 1: to which the following are now 
added to exercise the student. 


Ex. 1. Find the equation to the cissoid of Diocles ; whose 
manner of description is as below. 
From any two points Pp, s, at equal 
distances from the extremities a, B, of 
the diameter of a semicircle, draw st, 
eM, perpendicular to az, From the. 
point r where st cuts the semicircle, 
draw a right line ar, it will cut pm in) = 
M, a point of the curve required. Spee Bu se 
Now, by theor. 87 Geom. as . ss = sr? ; and by the con- 
struction, a8.sze=ap.ps. Also the similar triangles 4PM, 
Root eas CONSE 
AP 


Vou. IU. 38 quently 





» | 


AST, QIVE AP : PM 3: AS OST 2: PB: ST = 





296 EQUATIONS TO CURVES. 


rome pM? 2 2 pp . 
quently st? : ==AP.PB and lastly AP . AP?, 

or pa? =ps.pm?. Hence if the diameter an = d, ap = =a 
pm=y; the equation is x3=y* (d—2). | 

The complete cissoid will have another branch equal and 
similar to ama, but turned contrary ways ; being drawn by 
means of points T falling in the other half of the circle. But 
the same equation will comprehend both branches of the 
curve ; because the square of — y, as well as that of + 4, is 
positive. | 

Cor. All cissoids are similar figures ; because the abscissz 
and ordinates of several cissoids will be in the same ratio, when 
either of them 1s im a given ratio to the diameter of its gene- 
FAnne circle. 

| Ex. 2. Find the equation to the logarithmic curve whose 
fundamental property is, that when the abscissas increase or 
decrease in arithmetical progression, the corresponding ordi- 
nates increase or decrease in geometrical progression. 

Ans. y=a*, a being the number whose logarithm is 1, in 
the system of logarithms represented by the curve. 

Ex. 3. Find the equation to the curve called the Witch, 
whose construction is this: a semicircle whose diameter is AB 
being given ; draw, from any point pin the diameter, a per- 
pendicular OL AIALG, CUM INE the semicircle in p, and terminat- 
ing in M, so that ap: Pp: : ab: uM; thenis M always a point 

‘inthe curve. : ad— ro 
Ans, y=dy/— 





PROBLEM It... 


Given “the Equation to a Carve, to Describe it, and trace its 
Chief Properties. 


The method of effecting this is obvious : for any abscissas 
being assumed, the corresponding values of the ordinates be- 
come known from the equation ; and thus the curve may be 
traced, and its limits and properties developed. 

Fx. 1. Let the equation y*==a?x, or y=4{/a?r to a line of 
the third order be proposed. 

First, drawing the two indefinite lines 
BH, Dc, to make an angle Bac equal to 
the assumed angle of the co-ordinates ; 
let the values of « be taken upon ac, 
and those of y upon as, or upon lines’ 
parallel to as. Then, let it be enquired 
whether the curve passes through the 
point 4, or not. In order to this, we 
must aeveteh what y will be when 





EQUATIONS TO CURVES. 291 


x==0: and in that case y=%/(a? XO), that is, yan: There- 
fore the curve passes. through a. Let it next be ascertained 
whether the curve cuts the axis acin any other point; in or- 
der to which,. find the value of x when y = 0: this will be 
2/a* x=0, or  =.0. Consequently the curve does not cut 
the axis in any other point than a. Make x = ar = 1a, and 
the given equa. will become y = 2/1a3 = a3/1. Therefore 
draw pm parallel to ag and equal to a%/3, so will m.be a point 
in the curve. Again, make «= ac =a; then the equation 
will give y=3/a3=o. Hence, drawing oN parallel to as, and 
equal to ac or a, Nn will be another point in the curve. And 
by assuming other values of y, other ordinates, and conse- 
quently other points of the curve, may be obtained. Once 
more, making x infinite, or x = ©, we shall have y= 3/(a? 
Xoo); that is, y is infinite when x is so; and therefore the 
curve passes on to infinity. And further, since when ris 
taken = 0, it is ulso y=0, and when x = ©, it is also y=00 ; 
the curve will have no asymptotes that are parallel to the co- 
ordinates. 

Let the right line an be drawn to cut pm (produced if ne- 
cessary) ins. Then because cn= ac, it will be ps=ap=1n. 
But pPM=ai/i=1a7/4, which is manifestly greater than la: & 
so that pm is greater than Ps, and ppeegneury the curve is 
concave to the axis ac. 

Now, because in the given equation yp =a?x the exponent 
of xis odd, when xis taken negatively or on the other side 
of a, its sign should be changed, and the reduced equation 
will then be y =%/—a?x. Here it is evident that, when the 
values of x are taken in the negative way from 4 towards D, 
but equal to those already taken the positive way, there will 
result as many negative values of y, to fall below ap, and each 
equal to the corresponding values of y, taken above ac. Hence 
it follows that. the branch am’y’ will be similar and equal to the 
branch amy ; but contrarily posited. 

ix. 2. Let the lemniscate be proposed, which is a line of 
the fourth order, denoted by the equation a? 4? =a? x2 —z'. 


In this equation we have yata/(a? x2) 3) ATT 


where, when x = 0, y = 0, therefore the curve | |p!—'nj 
passes through a, the point from which the va- | 
lues of x are measured. Whenx== +a,then 7% 


y==0; therefore the curve passes throughs  » t 
and c, supposing ap and ac each = zh a. If a O 
were assumed greater than a, the value of y 3 | 


would become imaginary ; therefore no part of 3) 
the curve lies beyond sp or c. Whenz = ta, 


$92 EQUATIONS TO CURVES. 


then y= 1,/a — ta? = 1a 3; which is the value of the 
semi-ordinate pm when aP = Lag. And thus, by assuming 
other values of x, other values of y may be ascertained, and 
the curve described. It has obviously two equal and similar 
parts, and a double point ata. A right line may cut this 
curve in either 2 points, or in4: even the right line Bac is 
conceived to cut it in 4 points ; because the double point a is 
that in which two branches of the curve, viz. map, and na@, 
are intersected. 


Ex. 3. Let there be proposed the Conchoid of the ancients, 
which is a line of the fourth order defined by the equation 


(a? —2?) . (a —b)? =a y?, or yet ye —2z?). 


Here, if 2==0, then y becomes in- 
finite ; and therefore the ordinate at 
a (the origin of the abscissas) is an 
asymptote to the curve. If an = 6, 
and rv be taken between 4 ands, then 
shall pm and pm be equal, and lie on 
different sides of the abscissaar. If 
x==b, then the two values of y vanish, 
because «—b = 0, and consequently 
the curve passes through s, having 
there adouble point. If ar be taken 
greater than as, then will there be 
two values of y, as before having contrary signs ; that value 
which was positive before being now negative, and vice versa. 
But if ap be.taken = a, and p comes to np, then the two va- 
lues of y vanish, because in that case We —x?)=0. Ifar 
be taken greater than ap or a, then a? — x? becomes negative, 
and the value of y impossible : so that the curve does not go 
beyond pb. 

Now let z be considered as negative, or as lying on the 





side of a towards c. Then pace / (a? — 22). Here - 


if z vanish, both these values of y become infinite ; and con- 
sequently the curve has two indefinite arcs on each side the 
asymptote or directrix ay. If x increase, y manifestly dimi- 
nishes ; and. when a—=a, then y vanishes: thatis, if ac = an, 
then one branch of the curve passes through c, while the - 
other passes through p. Here also, if « be taken greater 
than a, y becomes imaginary ; so that no part of the curve 
can be found beyond c. 

If a=6, the curve will have a cusp ins, the node between 
B and D vanishing i in that case. Ifabe less than 6, then B will- 
become a conjugate point. Th 


EQUATIONS TO CURVES. ~ 293 


In the figure, mcm’ represents what is termed the superior 
eonchoid, and cpmpmsm the inferior conchoid. The point 
B is called the pole of the conchoid ; and the curve may be 
readily constructed by radial lines from this point, by means 


of the polar equation 2 Most +a. It will merely be re- 


quisite to set off from any assumed point Ay the distance spb ; 
then to draw through B a right line mim’ making any angle 
with cs, and from x the point, where this line cuts the direct- 
rix ay (drawn perpendicular to cz) set off upon it rm = Lm 
=a ; so shall mw’ and m be points in the superior and inferior 
conchoids respectively. 

Ez. 4, Let the principal properties of the curve whose 
equation is yx"=a"t?, be sought; when v is an odd number, 
and when n is an even number. 

Ex. 5. Describe the line which is defined by the equation 
xy--ay--cy=be-+ bz. 

Ex. 6. Let the Cardioide, whose equation is y4 —6ay? +- 
(2x2-+- 120?) y? — (6ax? +843) y+ (a? +3a2) a? =0, be pro- 
posed. 

Ex. 7. Let the Trident, whose equation is xyac? +bx2 + 
cx-+-d, be proposed. . 

Ex. 8. Ascertain whether the Cissoid and the Witch whose 
equations are found in the preceding problem, have asymptotes. 


PROBLEM If. 


To determine the Equation to any proposed Curve surface. 


_ Here the required equation must be deduced from the law 
or manner of constructions of the proposed surface, the re- 
ference being to three co-ordinates, commonly rectangular ones, 
the variable quantities being x, y, and z. Of these, twe, 
namely, « and y, will be foand in one plane, and the third Z 
will always mark the distance from that plane. 

Ex. 1. Let the proposed surface be that of a soheee, FNG. 

The position of the fixed point a, 
which is the origin of the co-ordinates 
ap, PM, MN, being arbitrary ; let it be 
supposed, for thegreater convenience, 
that it is at the centre of the sphere. | 
Let mA, Na, be drawn, of which the : 
latter is manifestly equal to the radius a ies 
of the sphere, and may be denoted by r. Then, if ar = 2, 





PM == Y, MN =z; the right-angled triangle arm will give 


AM* 


294 EQUATIONS TO CURVE SURFACES. 


am? =ap?- pm? =2?-+-y?. In like manner, the right-angled 
triangle AMN, posited in a plane perpendicular to the former, 
will give an? ==am? --mn?, that is, r? =a? +-y?2--z?, or 22=r2 
a? —y?, the equation to the ‘spherical surface, as required. 


Scholium. Curve surfaces, as well as plane curves, are 
Sinced in orders according to the dimensions of the equa- 
tions, by which they are represented. And in order to de- 
termine the properties of curve surfaces, processes must be 
em;ioyed, similar to those adopted when investigating the 
properties of plane curves.. ‘Thus, in like manner as in the 
theory of curve lines, the supposition that the ordinate y is 
equal to 0, gives the point or points where the curve cuts its 
axis ; so, with regard to curve surfaces the supposition of 
2 =0, will give the equation of the curve made by the in- 
tersection of the surface and its base, or the plane of the co- 
ordinates x, y. Hence, in the equation to the spherical sur- 
face, when z2 = 0, we have x?-+-y? = r?, which is that of a 
circle whose radius i is equal to that of the sphere. See A 
534 vol. 1. 

Ex. 2. Let the curve surface proposed be that produced 
by a parabola turning about its axis. 

_ Here the abscissas x being reckoned from the vertex or 
summit of the axis and on a plane passing through that axis ; 
the two other co-ordinates being, as before, y and z ; and the 
parameter of the generating parabola being p the equation of 
the parabolic surface will be found to be z2 -+y? —px=0. 


Now, in this equation, if z be supposed = 0, we shall have 
y? =px, which (pa. 534 vol. 1) is the equation to the generat- 
ing parabola, as it ought to be.. If we wished to know what 
would be the curve resulting from a section parallel to that 
which coincides with the axis, and at the distance a from it, 
we must put z=a; this would give y?=px—a?, which is still 
an equation toa parabola, but in which the origin of the ab- 
Scissas is distant from the vertex psa assumed by the quan- 


a2 


tity — ars 


fe By Ruppose the curve surface of a right cone were 
proposed. 

- Here we may most ConveRieaily refer the equation of the 
surface to the plane of the circular base of the cone. In this 
ease, the perpendicular distance of any point in the surface 
from the base, will be to the axis of the cone, as the distance 
of the foot. of that perpendicular from the circumference 

| (measured 


EQUATIONS TO CURVE SURFACES. 295° 


{measured on a radius), to the radius of the base: that is, if 
the values of x be estimated from the centre of the base, and 
* be the radius, z will vary as r—,/ (x3-++-y?): Conse- 
quently, the simplest equation of the conic surface, will be 
Zr ef CY? ), OF 73 — Breet ce gay 

Now from this the nature of curves formed by planes cut- 
ting the cone in different directions, may readily be inferred. 
Let it be supposed, first, that the cutting plane is inclined to 
the base of a right-angled cone in the angle of 45°, and passes 
through its centre: then will z2==«, and this value of z sub- 
stituted for it in the equation of the surface, will give r? — 
2rx=y" , which is the equation of the projection of the curve 
on the plane of the cone’s base: and this (art. 3 of this chap.) 
is manifestly an equation to a parabola. | 


Or, taking the thing more generally, let it be supposed that 
the cutting plane is so situated, that the ratio of x to z shall 
be that of 1 to m: then will mz =z, and m? 22=22. These 
substituted for z and z? in the equation of the surface, will 
give, for the equation of the projection of the section on the 
plane of the base, r? — 2mx-+-(m? — 1) 22? =y?. Now this 
equation, if m be greater than unity, or if the cutting plane 
pass between the vertex of the cone and the parabolic sec- ~ 
tion, will be that of an hyperbola: and if, on the contrary, the 
cutting plane pass between the parabola and the base, i. e. if 
m be less than unity, the term (m? — 1)x? will be negative, 
when the equation, will obviously designate any ellipse. 


Schol. It might here be demonstrated, in a nearly similar 
manner, that every surface formed by the rotation of any 
conic section on one of its axes, being cut by any plane what- 
ever, will always give a conic section. For the equation of 
such surface will not contain any power of x, y, or z, greater 
than the second ; and therefore the substitution of any values’ 
of z in terms of < or of y, will never produce any powers of 
x or of y exceeding the square. The section therefore must 
be a line of the second order. See on this subject, Hutton’s 
Mensuration, part ili, sect. 4. i ! 


Ex. 3. Let.the equation to the curve surface be xyz=a?. 


Then will the curve surface bear the same relation to the 
solid right angle, which the curve line whose equation is xy 
=a? bears to the plane right angle. ‘That is, the curve sur- 
face will be posited between the three rectangular faces 
bounding such solid right angle, in the same manner as the 

- equilateral hyperbola is posited between its rectangular asymp- 
totes. And in like manner as there may be 4 equal equila- 
| teral 


298 CONSTRUCTION OF EQUATIONS. 


teral hyperbolas comprehended between the same rectangular 
asymptotes, when preduced both ways from the angular point; 
so there may be 8 equal hyperboloids posited within the 8 
solid right angles which meet at the same summit, and all 
placed between the same three asymptotic planes. 


er 
SECTION II. 
On the Construction of Equations. 
PROBLEM I. 


To Construct Simple Equations, Geometrically. 


Here the sole art consists in resolving the fractions, to 
which the unknown quantity is equal, into proportional terms ; 
and then constructing the respective proportions, by means of 
probs. 8, 9, 10, and 87 Geometry. A few simple ate 
will render the method obvious. 


b 
1. Let r=; then c:a::6: 2. Whence a may be 
found by constructing according to preb. 9 Geometry. 


OAM eet First construct the proportion d : a : bs 


CP Which ath term call g; then a= ,orere::g em. 


3. Let z= —. Then, since a2 ena =(a+b) xX en . 


it will pete be necessary to construct the proportion ¢ oe 
+b::a—6: 2. 


st 
€ 





a?b— bc? ne : 

4. Let g== —— aaa Find, as in the first case, g= ake sdk 
a2b be bc? he 

pele — at === eee 1 { 

ai and A= 7? 80 that “7 May hen find by the first 


case = 2 i So shall a=g—7, the difference of those lines, 


found by construction. 








a2b— bad cig easy } ; 
5. Let a= afcbee First tind + the fourth proportion- 
al to b, a, and f, which make == h.. Then i); OR, 


by construction it wi behte:a —d::a: 2. 
; a? 
6, Let a= 





» Make the right-angled triangle ABC : such 
“that 


CONSTRUCTION OF EQUATIONS. 297 


that the leg an = a, Bc = 6; thenac=/ (4B? 
sc?) = Vf (aab?), by th. 34 Geom. “icnce cs 
Rs 
gt ee seonecract therefore the proportion 
c 


€:ac::ac:a, and the unknown quantity wil BoA 
be found, as required. . 


7, Let x = “ First, find cp a Fe 
L Rg & 
mean proportional between ac=c, and D —\> 


cp =d, that is, find co=/cd. Then - 
make cz =a, and join pe, which will \ 
evidently be = ,/ (a? + cd). Next on AC ES 
any line Ee set i pr==h-+-c. r¢ == ED; and draw au parallel 
to Fp, to meet pe (produced if need be) inu. So shall eu be 
=2, the third Berencas to h--e, and ,/ cere’), as re- 
uired. 
; Note. Other methods suitable to different cases which may 
arise are left to the student’s invention. And in all construt- 
tions the accuracy of the results, will increase with the size 
of the diagrams ; within convenient limits for operation. 


PROBLEM i. 


To Find the roots of Quadratic Equations by Construction. 


In most of the methods commonly giv- 
en for the construction of quadratics, it 
is required to set off the square root of 
the last term ; an operation which can . 
only be performed accurately when that 
term isa rational square. We shall here 
describe a method which, at the same 
time that it is very simple in practice, 
has the advantage of showing clearly the | 
relations of the roots, and of dividing the third term into two 
factors, one of which as least may be a whole number. _ 

In order to this construction, all quadratics may be classed 
under 4 forms : viz. 





1. 2?--ae —bc=0. 
2. x2 —axc—bc=0. 
St ee AE 


4, 22—axr+be=0. 

. One general ete of construction will toalude the first 
op ‘of these forms. Let x? yx —be==0, and 4 greater than 
-¢. Describe any circle asp having its diameter not less than 
the given quantities a and b—c, and within this circle inscribe 

Notas: 39 two 


298 CONSTRUCTION OF EQUATIONS. 


two chords as=a, an==b—c, both frem any common assumed 
point a. Then produce ap to F so that pr=c, and about the 
ceutre c of the former circle, with the radius cr, describe an- 
other circle, cutting the chords ap, as, produced in F, ©, G,H: 
so shall ac be the affirmative and au the negative root of the 
equation «?-+02—bc=0; and contrariwise ac will be the ne- 
gative and au the affirmative root of the equation «? —ax—be 
=—=,(), 

For, a¥ or aD-+ pr =6,and pF or aE=c ; and, making ac or 
Bu= «x, we shall have an==a-+x : and by the property of the 
circle ecru (theor. 61, Geom.) the rectangle Ea. ar—==Ga. an, 
or bc = (a-+2) x, or again by transposition x? +axz—- be == 0. 
Also if an be ==—2z, we shall have ac or BH Or AH—AB= —X 
—qa: and conseq. Ga. AH ==22-+ az, as before. So that, 
whether ac be = 2, or an =—za, we shall always have x?-+- 
ax—bc==0. And by an exactly similar process it may be 
proved that ac is the negative, and an the positive root of . 
x? — ax be=0. | 

Cor. In quadratics of the form z?-+-ax—bc=0, the positive 
root is always /ess than the negative root ; and in those of the 
form x? — az —bc=0, the positive root is always greater than 
the negative one. 


2. The third and fourth cases also are oar. 
comprehended under one method of con- 
struction, with two concentric circles. Let 
x2 + ax -+ be =: 0. Here describe any 
circle app, whose diameter is not less than H/ 
either of the given quantities a and b-+-c; _\/ 
and within that circle inscribe two chords 
AB =a, ap=b6-+c, both from the same 
point a. Then in ap assume prc, and about c the centre 
of the circle agp, with the radius cr describe a circle, cutting 
the chords ap, as, in the points F, Fr, G,H: so shall ac, an, 
be the two positive roots of the equation 2? —az-+-bc=0, 
and the two negative roots of the equation «?-+-ar-+-be=0. 
The demonstration of this also is similar to that of the first 
case. 

Cor. 1. If the circle whose radius is cr just touches the 
chord as, the quadratic will have two equal roots which can 
only happen when 1a?==bc. 

Cor. 2. If that circle neither cut nor touch the chord as, 
the roots of the equation will be imaginary : and this will 
always happen, in these two forms, when be is greater 
than 2a?. alae 






wh, 


PROBLEM 


CONSTRUCTION OF CUBICS, &c. 299 
PROBLEM. III. 


To Find the Root: of Cubic and Biquadratic Equations by 
Construction. 


1. In finding the roots of any equation, containing only one 
unknown quantity, by construction, the contrivance consists 
chiefly in bringing a new unknown,quantity into that equation ; 
so that various equations may be had, each containing the two 
unknown quantities ; and further, such that any two of them 
contain togetier all the known quantities of the proposed 
“equation. Then from among these equations two of the most 
simple are selected, and their corresponding loci constructed ; 
the intersection of those loci will give the roots sought. 

Thus it will be found that cubics may be constructed by 
two parabolas, or by a circle and a parabola, or by a circle 
and an equilateral hyperbola, or by a circle and an ellipse, 
&c.: and biquadratics by a circle and a parabola, or by a cir- 
cle and an ellipse, or by a circle and an hyperbola, &c. Now, 
since a parabola of given parameter may be easily construct- 
ed by the rule in cor. 2 th 4 Parabola, we select the circle 
and the parabola, for the construction of both biquadratic and 
cubic equations. The general methed applicable to both, 
will be evident from the following description. 

2. Let m” am’ be a parabola whose 

axis is AP, M’m’am acircle whose cen- 
tre is c and radius cm, cutting the pa- 
rabolawn the points um, m,m’', mw”: from 
these points draw the ordinates to the 
axis MP, M’P’, M’e”, me” : and from c 
let fall cp perpendicularly to the axis : 
also draw cn parallel to the axis: meet- 
ing pMinN. Let ap==a, pc= 6, cm=<n, 
the parameter of the parabola = == 2, 
ap=x, pM=y. Then (pa. 534 vol. 1) px = y? : also cm? = 

cn? -nM?, or 1? = (a4ra)?-+-(y52b)? ; that is, 2? + 2ax + a3 
+y ee Reta a Substituting in this equation for z, its 





value y and arranging the terms accerding to the dimensions 


of y, there will arise - 

y* £(2pa-tp? )y? © 2bp2y+ (a? +b? —n? )p?=0, 
a biquadratic equation whose roots will be expressed by the 
ordinates pm, p/m’, p’m’, Pn’, at the points of intersection of 
the 5 BOP parabola and Fale 

To make this coincide with any Brenoned biquadratic 
re second term is taken <¢ away (by cor. theor. 3); assume. 

hy =e 


300 CONSTRUCTION OF CUBICS, &c. 


yt+—qy?-+ry — s=0. Assume also p = 1; then comparing 
the terms of the two equations, it will be, 2a—1=q, or a= 

7 | gine | 
- — 2h==r, or b= > ; a? +62 —n? = — s, or n? =a? -- 
Bg i and consequently n= ,/(a?+b2+s). Therefore de- 
scribe a parabola whose parameter is 1, and in the axis take 


‘Bik 


(ae 





: at right angles- to it draw pc and = —1ir; from 


the sevice c, with the radius a (02° 4. DF rt Ns describe the 
circle m’m’em, cutting the parabola 1 in the points m, Mm’, 1”, m’” ; 
then the ordinates pm, Pm’, Pm”, pe’, will be the rootasre-*> 
quired. 

Note. This method, of making p = 1, has the obvious ad- 
vantage of requiring only one parabola for any number of 
biquadratics, the necessary variation being made in the radius 
of the circle. 


Cor. 1. When vc represents a negative quantity, the ordi- 
nates on the same side of the axis with ¢ represent the nega- 
tive roots of the equation ; and the contrary. 


Cor. 2. If the circle touch the parabola, two roots of the 
equation are equal ; if it cut it only im two points, or touch it 
in one, two roots are impossible ; andif the circle fall wholly 
within the parabola, all the roots are impossible. 

Cor. 3. If a2 -+ 6? =n?, or the circle pass through the 
point’a, the last term of the equation, 1.€. a2--+-b? —n? ea 
and therefore ie PULP )y? + 2bp?2y==0, or 
yt (2pa-+ p? jy & 2bp? == 0. This cubic equation may be 
mnade to coincide with any proposed cubic, wanting its second 
term, and the ordinates pm, em", Pm”, are its roots. 

Thus, if the cubic be expressed “pennealty by y? + gyts=—0. 
By comparing the terms of this and the preceding equation, 
we shall have + 2pa +- p? = + gq, and + 2bp? = + 5, or 


+-a=1lp>= i. and === =e So that, to construct a cubic 


equation, with any given parabola, whose half parameter is 
aB (see the preceding figure) ; from the point g take in the 
axis, (forward if the equation have =i but backward if q be 
positive ) ) the line pps, then raise the perpendicular pe 
= = and from c describe a circle passing through the ver- 
tex a of the ‘parabola; the ordinates pm, &c. drawn from the 
points of intersection of the circle and parabola, will be the 


roots required. 
PROBLEM 


CONSTRUCTION OF CUBICS. «> SGa 


PROBLEM IY. 

To Construct an Equation of any Order by means of a Locus 
of the same Degree as the Equation proposed, and a Right 
Line. 

As the general method is £ 
the same in all equations, let = #}--..- 
it be one of the 5th degree, as us cinM7 NR dA 
25 —bx4 +acxr? —a2 dx? +a3 | fe bak Nuk 
ex—a4f=0. Let the last term 
af be transposed ;_ and, tak- RB 
ing one of the linear divisors, _ 7 
f, of the last term, make it : 
equal to z, for example, and divide the equation by a+ ; then 
¢> —bx* Lace? —a2dxrt-+atex 

— ' 

‘On the indefinite line gq describe the curve of this equa- 
tion, BMDRLFC, by the method taught in prob. 2, sect. 1, of 
this chapter, taking the values of « from the fixed points. 
The ordinates pm, sr, &c. will be equal to z ; and therefore, 
from the point g draw the right line sa = f, parallel to the 
ordinates pm, sr, and through the point a draw the inde- 
finite right line xc both ways, and parallel to ze. From the 
points in which it cuts the curve, let fall the perpendiculars, 
mp, RS, cq: they will determine the abscissas BP, BS, Be, 
which are the roots of the equation proposed. ‘Those from 
a towards @ are positive, and those lying the contrary way are 
negative. 

If the right line ac touch the curve in any point, the cor- 
responding abscissa z, will denote two equal roots ; and if it 
do not meet the curve at all, all the roots will be imaginary. 

If the sign of the last term, a*f, had been positiye, ee 
we must have made z =—/, and therefore must have taken 
ba==—/f, that is, below the point e, or on the negative side. 

EXERCISES. 

_Ex. 1. Let it be proposed to divide a given arc of a circle 
into three equal parts. 

Suppose the radius of the circle to be represented by a 
the sine of the given arc by a, the unknown sine of its third 
part by a, and let the knows: arc, be 3u, and of course, the 
Ps arcbeu. Then, by equa. viit, 1x, chap. iil, we shall 

ave 










will z= 


: sin 2a. cos u--cos eu S 
sin 3u==sin (Quu) = a0 Ate SOs eee, ee - sin ue 


: Sees a 2 sinw. cos 
sin. eee ( uu) a nea 
r 
‘cos! u—sin? wu 
cos 2u=008 ( ey cE SRR aE, A ese ae 
fate e Putting’ 








302 TRISECTION OF AN ARCH. ~ 


Putting, in the first of these equations, for sin Su its given 
value a, and for sin 2u, cos 2u, their values given in the te 

other equations, there will arise 
__3sin uw. cos? uw. sin? u 














r 
Then substituting for sin wu its value x, and for cos? w its va- 
lue r? — x2 and arranging all the terms according to the Pot 
ers of x, we shall have 
£3 mm 273 y-+-lar? =O, 

a gable equaliag of the form x3 —px-+-q=0, with the condition 
that ..p? > 19? ;, that is to say, itis a cubic equation falling 
under the irreducible case, and its three roots are represent- 
ed by the sines of the three arcs u, wu -- 120°, and u-+4-240°. 

Now, this cubic may evidently be constructed by the rule 
in prob. 3 cor 3. But the trisection of an arc may also be 
effected by means of an equilateral hyperbola, in the follow- 
ing manner. 


Let the arc to be trisected be as. 

In the circle asc draw the semi- 
diameter ap, and to ap as a diame- 
ter, and to the vertex a, draw the 
_ equilateral hyperbola az to which 
the right line ap (the chord of the 
arc to be trisected) shall be a tangent in the point a; then 
the arc ar, included within this hyperbola, i is one third of the 
arc AB. 
For, draw the chord of the arc ag, bisect ap at c, so that 
g will be the centre of the hyperbola, join pr, and draw cu 
parallel to it, cutting the chords az, ar, init and Kx, .Then, 
the hyperbola being equilateral, or having its transverse and 
conjugate equal to one another, it follows from Def. 16 Conic 
Sections, that every diameter is equal to its parameter, and 
from cor. theor, 2 Hyperbola, that cx x1 = ax?, or that 
GK : AK:: AK : KI; therefore the triangles Gxa, aki are 
similar, and the angle kar = ack, which is manifestly = apr. 
Now the angle apr at the centre of the circle being equal to 
KAI or FAB; and the former angle at the centre being mea- 
sured by the arc ar, while the latter at the circumference is 
measured by half re; it follows that ar—trsB, or = 1 ap, as 
it ought to be. 

Ex. 2. Given the side éf a cube, to find the side of another 
of double capacity. 

Let the side of the given cube be a, and that of a double 
one y, then 2a3=y3, or by putting 2u=b, it will be a? =by?; 
there are therefore to be found two mean ee be- 

tween 





DUPLICATION OF THE. CUBE. 303 


tween the side of the cube and twice that side, and the first 
of those mean proportionals will be the side of the double 
cube. Now these may be readily found by means of two pa- 
rabolas ; thus : | 


Let the right lines ar, as, be joined 
at right angles ; and a parabola amu be 
described about the axis ar, with the 
parameter a; and another parabola amr 
about the axis as, wth the parameter b : 
cutting the former in mM. Then ap==z2, 
pmM==y, are the two mean proportionals 
of which y is the side of the double cube required. 





For, in the parabola amu the equation is y? az, and in the 
parabola amr itis <2 ==by. Consequently a: y:: y: 2, and 
y:a2::a:6. Whence ye =ab; or, by substitution, y ./ by 
—=ab, or by squaring y?b==a?b? ; or lastly, y®=a?b= 2a, as 
it ought tobe. 

; a 


THE 


-f 304 | 


_ ‘THE DOCTRINE OF FLUXIONS. — 
DEFINITIONS AND PRINCIPLES. 


Jirt. 1. In the Doctrine of Fluxions, magnitudes or quan- 
tities of all kinds are considered, not as made up of anumber 
of small parts, but. as generated. by continued motion, by 
means of which they increase or decrease. As, a line by the 
motion of a point; a surface by the motion of a line; and a 
solid by the motion of a surface. So likewise, time may be 
considered as represented by a line, increasing uniformly by 
the motion of a point. And quantities of all kinds whatever, 
which are capable of increase and decreasegmay in like man- 
ner be represented by geometrial magnitudes, conceived to 
be generated by motion. 


2. Any quantity thus generated, and variable, is called a 
Fluent, or a Flowing Quantity. And the rate or proportion 
according to which any flowing quantity increases, at any po- 
sition or ‘instant, is the Fluxion of the said quantity at that po- 
sition or instant: and it is proportional to the magnitude by 
which the flowing quantity would be uniformly increased in a 
given time, with the generating celerity uniformly continued 
during that time. 


3. The small quantities that are actually generated, ‘/pro- 
duced, or described, in any small given time, and by any con- 
tinued motion either uniform or variable, are called Incre- 
ments. 


4, Hence, if the motion of increase aie uniform, by Shick 
increments are generated, the increments will in that case be 
proportional, or equal, to the measures of the fluxions : but: 
if the motion of increase be accelerated, the increment so 
generated, in a given finite time, will exceed the fluxion: 
and if it be a decreasing motion, the increment, so generated, 
will be less than the fluxion. But if the time be indefinitely 
small, so that the motion be considered as uniform for that 
instant; then these nascent increments will always be pro- 
portional, or equal, to the fluxions, and may be substituted 
neta of them, in any calculation. ; 

| ; 5° To 


FLUXIONS. 308 


5. To illustrate these definitions: Sup- Mm 
pose a point m be conceived to move from 4 ‘a 
the position a and to generate'a line ap, 
by a motion any how regulated ; and sup- 
pose the celerity of the point m, at any position P, to be such, 
as would, if from thence it should become or continue uni- 
form, be ‘sufficient to cause the point to describe, or pass uni- 
formly over, the distance rp, in the given time allowed for 
the fluxion : then will the said line pp represent the fluxion 
of the fluent, or flowing line, ap, at that position. 





6. Again, suppose the right 
line mn to move, from the posi- 
tion 4B, continually parallel to 
itself,with any continued motion, 
so as to generate the fluent or 
flowing rectangle asep, while 
the point m describes the line 
AP: also, let the distance pp be taken, as before, to express 
the fluxion of the line or base ap; and complete the rectan- 
gle pagp. Then, like as pp is the fluxion of the line ap, so 
is pq, the fluxion of the flowing parallelogram ag : both these 
fluxions, or increments, being uniformly described in the same 
time. 





7. In like manner, if the solid 
aERP be conceived to be gene- 
rated by the plane rer, moving 
from the position ase, always pa- 
rallel to itself, along the line ap ; 
and if pp denote the fluxion of 
the line ae: Then, like as the 
rectangle pagp, or pa X Pp, denotes the fluxion of the flowing 
rectangle ABQP, so also shall the fluxion of the variable solid, 
or prism aperep, be denoted by the prism Paxrgp, or the 
plane rR X pp. And, in both the last two cases, it appears 
that the fluxion of the generated rectangle, or prism, is equal 
to the product of the generating line, or plane drawn into the 
fluxion of the line along which it moves. 





8. Hitherto the generating line, or plane, has been consi- 
dered as of a constant and invariable magnitude ; in which 
case the fluent, or quantity generated, is a rectangle, or a 
prism, the former being described by the motion of a line, 
and the latter by the motion of a plane. So, in like manner 
are other figures, whether plane or solid, conceived to be de- 

v Vor. IF; 40 scribed 


306 DEFINITIONS. 


scribed: by the motion of a Variable Magnitude, whether it 
be a line or a plane. Thus, let a variable line rq be carried 
_ by a parallel motion along ap; or while a point P is carried 
along, and describes the line ap, suppose another point, @ to 








Es BAe eeaies 


be carried by a motion perpendicular to the former and to de- _ 
scribe the line re: let pg be another position of Pa, indefi- 
nitely near to the former ; and draw qr parallel to arp. Now 
in this case there are several fluents, or flowing quantities, 
with their respective fluxions : namely, the line or fluent ar, 
the fluxion of which is pp or gr; the line or fluent Pa, the 
fluxion of which is rq; the curve or oblique line aq, described 
by the oblique motion of the point @, the fluxion of which is 
aq; and lastly, the surface apa, described by the variable line 
rq, the fluxion of which is the rectangle parp, or Pa X Pp. 
In the same manner may any solid be conceived to be describ- 
ed, by the motion of a variable plane parallel to itself, sub- 
stituting the variable plane for the variable line; in which 
case the fluxion of the solid, at any position, is represented 
by the variable plane, at that position, drawn into the fluxion 
of the line along which it is carried. 


9. Hence then it follows in general, that the fluxion of any 
figure, whether plane or solid, at any position, is equal to the 
section of it, at that pesition, drawn into the fluxion of the 
axis, or line along which the variable section is supposed te 
be perpendicularly carried: that is, the fluxion of the figure 
AQP, is equal to the plane pa XPp, when that figure is a solfd, 
or to the ordinate pe X rp, when the figure is a surface. 


10. It also follows from the same premises, that in any 
eurve or oblique line 4g, whose absciss is ap, and ordinate Is 
pa, the fluxions of these three form a small right-angled 
plane triangle egr.; for ar = pp is the fluxion of the absciss 
aP, gr the fluxion of the ordinate rq, and eg the fluxion of 
the curve or right line ag. And consequently that, in any 
curve, the square of the fluxion of the curve, is equal to the 

sum 


= 


NOTATION. 307 


sum of the squares of the fluxions of the absciss and ordinate, 
when these two are at right angles to each other. 

11. From the premises it also appears, that contemporane- 
ous fluents, or quantities that flow or increase together, which 
are always in a constant ratio to each other, have their 
fluxions also in the same constant ratio, at every position. 
For, let ap and Be be two centempo- 
raneous fluents, described in the same 
time by the motion of the points pr and 
a, the contemporaneous positions be- 
ing P,Q, and p,q; andletar be tose, 5 
or ap to Bq, constantly in the ratio of 1 
to n. 

Then + - - - = ‘isnXap=se, 

‘ and 2 X ap=x@ ; 
therefore, by subtraction, n XPrp=aq; 
that is, the fluxion - pp:: fluxion eg: : 1: n, 
the same as the fluent ap : fluent pe :: 1:7, 
or, the fluxions and fluents are in the same constant ratio. 

But if the ratio of the fluents be variable, so will that of 
the fluxions be also, though not in the same variable ratio with 
the former, at every position. 


ty 
Ol 
a) 


NOTATION, &c. 


* 


12. To apply the foregoing principles to the determination 
of the fluxions of algebraic quantities, by means of which 
those of all other kinds are assigned, it will be necessary first 
to premise the notation commonly used in this science, with 
some observations. As, first, that the final letters of the 
alphabet z, y, x, u, &c. are used to denote variable or flowing 


quantities ; and the initial letters, a, 6, c,d, &c. to denote 


constant or invariable ones: Thus, the variable base ap of 
the flowing rectangular figure azap, in art. 6, may be repre- 
sented by x; and the invariable altitude ra, by a: also, the 
variable base or absciss ap, of the figures in art. 8, may be 
represented by «, the variable ordinate pe, by y; and the va- 
riable curve or line aq, by z. 
Secondly, that the fluxion of a quantity denoted by a single 
letter, is represented by the same letter with a point over it : 
Thus, the fluxion of « is expressed by x, the fluxion of y by 
y, and the fluxion of z by z. As to the fluxions of constant 
or invariable quantities, as of a, 6, c, &c. they are equal to 


_ nothing, because they do not flow or change their pee 


hirdly, 


308 DIRECT METHOD OF FLUXIONS. 


Thirdly, that the increments of variable or flowing quanti- 
ties, are also denoted by the same letters with a small’ over 
them: Thus, the increments of «, y, z, are x’, y', 2’. 

13. From these notations, and the foregoing principles, the 
quantities and their fluxions, there considered, will be denoted 
as below. Thus, in all the foregoing figures, put 


the variable or flowing lime - - ar=da, 
in art. 6, the constant line - - Pama, 
in art. 8, the variable ordinate - ra=y, 
also, the variable line or curve - aq=2z: 


Then shall the several fluxions be thus bepresented, namely, 

x = Pp the fluxion of the line ar, 

ac = Pegp the fluxion of anee in art. 6, 

yx = pgrp the fluxion of are in art. 8, 
z= ag=/ (x? +y7) the fluxion of ag; and 

dz = pr the fluxion of the solid in art..7, "if a denote the 

| constant generating plane PQR ; also, 
nx = BQ in the figure to art. 11, and 
na = aq the fluxion of the same. 


14. The principles and notation being now bid down, we 
may proceed to the practice and rules of this doctrine ; which 
consists of two principal parts, called the Direct and Inverse 
Method of Fluxions; namely, the direct method, which 
consists in finding the fluxion of any proposed fluent or flow- 
ing quantity ; and the inverse method, which consists in find- 
ing the fluent of any proposed fluxion. As to the former of 
these two problems, it can always be determined, and that in 
finite algebraic terms; but the latter, or finding of fluents, 
can only be effected in some certain cases, except by means 
of infinite series.—First then, of 


THE DIRECT METHOD OF FLUXIONS. 


Lo find the Fluxion of the Product. or Rectangle of two Varia- 
ble Quantities. 


15. Let arep, = xy, be the flow- 
ing or variable rectangle, generated 
by two lines pe and re, moving al- 
ways perpendicular to each other, 
from the positions.ar and ap; denot- 
ing the one by x and the other by y ; 

supposing « andy to be so related, 
that the curve line a@ may always 
pass through the intersection g of those lines, or the opposite 
angle of the rectangle. Now, 





DIRECT METHOD OF FLUXIONS. 309 


’ Now, the rectangle consists of the two trilinear Spaces arg, 
ara, of. which, the 

fluxion of the former is pa X Pp; or yx, 

that of the latter is - nqXnrr, or ry, by art. 8; 
therefore the sum of the two xy + xy, is the Hanion of the 
whole rectangle xy or aR@p. 


The Same Otherwise. 


16. Let the sides of the rectangle x and y, by flowing, be- 
come r+’ and y+y’: then the product of these two, or xy 
-+ay'+ya'+a'y’ will be the new or contemporaneous value of 
the flowing rectangle pr or xy: subtract the one value from 
the other, and the remainder, ry + yz’ . xy, will be the in- 
crement geherated i in the same time as x’ or y’; of which the 
last term 2’y’ is nothing or indefinitely small, in respect of the 
other two terms, because 2’ and 7 are indefinitely small in 
respect of x and y; which term being therefore omitted, there 
remains vy bya for the value of the increment ; and hence, 
by substituting x and y, for x’ and 7, to which they are pro- 
portional, there arises xy + yx for the true value of the flux- 
ion of xy; the same as before. 


17. Hence may be easily derived the fluxion of the pow- 
ers and products of any number of flowing or variable quan- 
tities whatever ; as of xyz, or uryz, or vuxyz,&c. And first, 
for the fluxion of ayz: put p=ay, and the whole given fluent 
ayz=q, or g=axyz=pz. Then, taking the fluxions of g=pz, 
by the last article, they are gy =fz + pz; but p= xy, and so 
p=xy+z2y by the same article : substituting therefore these 

values of p and f instead of them, in the value of g this be- 
comes g =xy2-+aryz+ xyz, the fluxion of xyz required ; which 
is therefore equal to the sum of the products, arising from the 
fluxion of each letter, or quantity, multiplied by the Bypdact 
of the other two. 


Again to determine the fluxion of wayz, the continual pro- 
duct of four variable quantities ; put this product, namely 
uxyz, or qu==r, where g=«yz as above. Then, taking the 
flaxions by the last article, r= gu-++ gu; which, by substitut- 
ing for q and g their values as above, becomes. - - - - 
r= = unye--uscye--ury2--uryz, the fluxion of uxyz as requir- 
ed: consisting of the fluxion of each quantity, drawn into the 
products of the other three. 

In 


310 DIRECT ‘METHOD OF FLUXIONS. 


In the very same manner it is found, that the fluxion of 
| vuxyz is puxye--ounyz--ouxry2 + vuryz ox vucyz ; and so on, 
for any number of quantities whatever ; im which it is always 
found, that there are as. many terms as there are variable 
quantities in the proposed fluent; and that these terms con- 
‘ist of the fluxion of each variable quantity, multiplied by 
the product of all the rest of the quantities. 


18. Hence is easily derived the fluxion of any power of a 
variable quantity, as of «?, or«%, or a*, &c. For, in the 
product or rectangle ay, if x =y, then is ry= «rz or x?, and 
also its fluxion xy-+-ay= xz-+rx or Yrx, the fluxion of x?. 


Again, if all the three x, y, z, be equal ; then is the product 
of the three xyz = ax? ; and consequently its fluxion xyz + 
ayzf-ayzsrertaxe trix or 3x?x, the fluxion of 23. 

In the same manner it will appear that 

the fluxion of 24 is == 4x37, and - 
the fluxion of 25 1s = 5x4 x and, in general, 
the fluxion of 2” is==nz"—!z; 

where 7 is any positive whole number whatever. 

That is, the fluxion of any positive integral power is equal 
to the Auxidc of the root (x), multiplied by the exponent of 
_the power (n), and by the power of the same root whose in- 
dex is less by 1, (x"—). 


And thus, the fluxion of a-+-cx being cx, 
that of (a ifn ex)? is 2ez7 X(a+cx ) or 2acr+2c%xrx, | 
that of (a +-ex?)? is 4crz X(a+cx?) or 4acta+4e7?z3 x, 
that of (72-++ 42)? is (4rz-+4yy ) X(x?+y?), 
that of (w --cy?)9 is (3x -+-6eyy )X(a +-cy?)?. 


19. From the conclusions in the same article, we may also 
derive the fluxion of any fraction, or the quotient of one va- 
riable quantity divided by another, as of 


y For, put the quotient or fraction — = q; then, multiply- 


ing by the denominator, c==qy ; and, taking the fluxions, 
r= gy+qy, or qy=x —qy ; and, by division, 


g Se (by substituting the value of q, or my 
te a Sasces Sata the fluxion of —, as Reacwel: 
A y y | 


That 


DIRECT METHOD OF FLUXIONS. 311 


That is the fluxion of any fraction, is equal to the fluxion 
of the numerator drawn into the denominator, minus the flux- 
ion of the denominator drawn into the numerator, and the re- 
mainder divided by the square of the denominator. 
nya Oy — ay 


9 


So that the fluxion of a is a x 
20. Hence too is easily derived the fluxion of any negative 

1 
integer power of a variable quantity, as of <~—”, or ee which 


is the same thing. For here the numerator of the fraction 
is 1, whose fluxion is nothing; and therefore, by the last 
article, the fluxion of such a fraction, or negative power, 
is barely equal to minus the fluxion of the denominator, 


divided by the square of the said denominator. That is the 
n—1_2 * nx , i 
= a or—nz—"—! »; or 





3 
fluxion of x—", or —is — 
be baal 


the fluxion of any negative integer power of a variable 
quantity as 2—”, is equal to the fluxion of the root, multiplied 
by the exponent of the power, and by the next power less by 
1; the same rule as for positive powers. 

‘The same thing is otherwise obtained thus: Put the pro- | 


1 
posed fraction, or quotient — -—==q; thenis 98 n= 1; and tak- 


ing the fluxions, we have 

g2"+gna"—1e=0, hence ness divide by x”, then 
g=- Ine (by substituting — — 1 for #), wat Fr ors—ne—"—! 5 
the ame as before. | 


1, x 
Hence the fluxion of z—! or —is—a—2x or — — 
Xx 


el 
ers evn Qa 
that of — - a—? or —is—2a-3z or -—, 
1 3x 
that of - a~S or —is-3a—47 Or ——, 
he x 
Aaze 
that of - ax—tor ~is—dax-f2 or, 
x 


that of (a++x)—! or ae is —(a-+-x)—2x or aay 


that semanas coe rare aya 38S — 12cxx x (a-+-322)- 3 


12cxr« 
(a3? )®’ 
21. Much in the same manner is obtained the fuxion of 
any 


OT == 


312 DIRECT METHOD OF FLUXIONS. 


™m 


any fractional power of a fluent quantity, -as of a”, or Chae. 


For, put the proposed quantity 9; then, raising each 
side to the n power, gives ae" 5 
taking the fluxions, gives ma"\y==ngr—) § g; then 

mam—|s. mx”—\ x m 4a 
dividing by ng"—', gives g = ————=——___=- xn 2. 
eg Sie 

- Which is still the same rule, as before, for finding the fluxion 
of any power of a fluent quantity, and which therefore is ge- 
neral, whether the exponent be positive or negatives integral 


or fractional. -And ‘hence the fluxion of oe? is 3 edoctbey 





| 1 ae p,' Be se. 
that of ax? is 3 ax?—1z =1ar—?z7 = ont 2/5 and that. of 
4/ (a? —x?) or (a2 —a2)2 18 }(a? —22)? X —2a= Sa ay" 


22. Having now found tit the fluxions of all SASH i y 
forms of algebraical quantities ; it remains to determine those 
of logarithmic expressions and also of exponential ones, that 
is such powers as have their exponents variable or flowing 
quantities. And first, for the fluxion of Napier’s, or the hy- 
perbolic logarithm. 

23. Now, to determine this from 
the nature of the hyperbolic spaces. 
Let a be the principle vertex of an 
hyperbola, having its asymptotes cp, 
cp, with the ordinates pa, BA, PQ, : 
&c. parallel to them. Then, from ¢ B PP 
the nature of the hyperbola and of ; 
logarithms, it is known, that any space appre is the log. of 
the ratio of cp to cp, to the modulus asco. Now, put t=cs 
or BA the side of the square or rhombus pg ; m= the modu- 
lus, or cB XBa; or area of ps, or sine of the angle c to the 
radius 1; also the absciss cp =x, and the ordinate ra=y. 
Then, by the nature of the hyperbola, cp X rq is always equal 





to ps, that is, cy=m: hence y= = and the fluxion of the 


space, ay is me ragp the fluxion of the log. of x, to the mo- 
dulus m.. And, in the hyperbolic logarithms, the modulus m 
being 1, therefore is the fluxion of the hyp. log. of x; 

which 


OF SECOND, THIRD, &c. FLUXIONS. 313 


which is therefore equal to the fluxion of the quantity, divided 
by the quantity itself. 
Hence the jum of the hyp. log. 


Ere iba? 


e es 
of 1 — zis ——-, 
1—z 


“pulaek 


of ee ie x(a—2)- +a(a+2) , a—r ax 

a—x eee Cer ee apie a? — x2" 
pie 
. ene : Sia 

24. By means of the Aikions of logarithms, are ‘usually 
determined those of exponential quantities, that is, quantities 
which have their exponent a flowing or variable letter. These 
exponentials are of two kinds, namely, when the root isa con- 
stant quantity, as e*, and when the root is variable as well as 
the exponent, as y* 

25. In the first case put the exponential, whose fluxion is 
to be found, equal to asingle variable quantity z, namely, z==ex ; 
then take the logarithm of each, so shall log. za X log. e ; 





of ax” is 


take the fluxions of these, so shall ~ pu X log. e, by the last 


article : hence z =zx Xlog. guic? log e, which is the flux- ~ 
ion of the proposed quantity e* or 2; and which therefore is 
equal to the said given quantity drawn into the ixtsion of the 
exponent, and into the log. of the root. 

Hence also, the fluxion of (a-+c)"* is (a-ee)n# X nx X log. 

a+c 

ek like manner, in the second case, put the given quan- 
tity y”==z; then the logarithms give log. z=x Xlog. y, and the 


fluxions give = 2 == x Xlog. yx xT ; hence z= zz X log. y+ 


ae (by substituting y* for z) y*x Xlog. y+ay*—ly, which is 


the fluxion of the proposed quantity yz ; and which therefore 
consists of two terms, of which the one is the fluxion of the 
given quantity considering the exponent as constant, and the 
other the fluxion of the same quantity considering the root as 
constant. 


Vor. Il. Oy es ist A . A Oe 


‘{ 314 J 
OF SECOND, THIRD, &c. FLUXIONS, 


Havine explained the manner of considering and determin- 
ing the tirst fluxions of flowing or variable quantities ; it re- 
mains now to consider those of the higher orders, as second, 
third, fourth, &c. fluxions. 

27. If the rate or celerity with which any flowing quantity 
changes its magnitude, be constant, or the same at every posi- 
tion ; then is the fluxion of it also constantly the same. But 
if the variation of magnitude be -continually changing, either 
increasing or decreasing ; then will there be acertain degree 
of fluxion peculiar to every point or position; and the rate of 
variation or change in the fluxion, is called the Fluxion of the 
Fluxion, or the Second Fluxion of the given fluent quantity. 
In like manner, the variation or fluxion of this second fluxion, 
is called the Third Fluxion of the first proposed fluent quan- 
tity ; and so on. 

These orders of fluxions are denoted by the same fluent 
letter with the corresponding number of points over it; 
namely, two points for the second fluxion, three points for 
the third fluxion, four points for the fourth fluxion, and so on. 
So, the different orders of the fluxion of x. are x, 2, 75 2, SC: 3 
where each is the fluxion of the one next before it. 

28. ‘This description of the higher orders of fluxions may 
be illustrated by the figures exhibited in art. 8, page 306 ; 
where, if « denote the absciss ar, and y the ordinate ra: and 
if the ordinate rq or y flow along the absciss ar or x, witha 
uniform motion; then the fluxion of x, namely, x=prp or gr, 
is a constant quantity, or x= 0, in all the figures. Also, in 
fig. 1, in which aq is a right line, y==rq, or the fluxion of re, 
is a constant quantity, or y= 0 ; for, the angle e, = the an- 
gle a, being constant, aris to rg, or x toy, Ina constant ra- 
tio. But in the 2d fig. rg, or the fluxion of pe, continually 
increases more and more; and in fig. Sit continually decreases 
more and more, and therefore in both these cases y has a se- 
cond: fluxion, being positive in fig 2, but negative in fig. 3. 
And so on, for the other orders of fluxions. 

Thus if, for instance, the nature of the curve be such, that 
2* is every where equal to a?y; then, taking the fluxions it is 
a?y = 3x2x ; and, considering x always as a constant quantity, 
and taking always the fluxions, the equations of the several 
orders of fluxions will be as below, viz. 

the ist fluxions a?y =3x? x, 

the 2d. fluxions a?y =6az?, 

the 3d. fluxions ata xb? : 

the 4th fluxions a? y=0, 

and all the higher fluxions also = 0, or nothing. 


OF SECOND, THIRD, &c. FLUXIONS. 315 


Also, the higher orders of fluxions are found in the same 
manner as the lower ones. . Thus, 
the first fluxion of y? is - .- - -  3y%y;5 
its 2d flux. or the flux. of 3y?y, con- | 
sidered as the rectangle of 3y?,.) 3y2y-+-6yy? : 
SH veres Bp nt 7! Se ae 
and the flux. of this again, or the 3d ‘ iy 
re of y®,is- Bosna £ 3y2 i 18y iy + By ®, 
29. In the foregoing articles, it has been supposed that the 
fluents increase, or that their fluxions are positive ; but it 
often happens that, some fluents decrease, and that therefore 
their fluxions are negative: and whenever this is the case, 
the sign of the fluxion must be changed, or made contrary to 
that of the fluent. So, of the rectangle zy, when both x and 
yincrease together, the fluxion is xy + xy; but if one of 
them, as y, decrease, while the other, x, increases ; then, the 
fluxion of y being — y, the fluxion of xy will in that case be 
axy—zy. This may be illustrated by 
the annexed rectangle, arpqgrn = «ry, 
supposed to be generated by the mo- 
tion of the line Pg from a towards c, 
and by the motion of the line re from 
8 towards a: For, by the motion of 
Pq, from a towards c, the rectangle 
is increased. and its fluxion is + xy ; 
but, by the motion of ne, from B to- 
wards a, the rectangle is decreased, 
and the fluxion of the decrease is xy; therefore, taking the 
fluxion of the decrease from that of the increase. the fluxion 
of the rectangle xy, when x increases and y decreases, is zy 
—Zy. 





REMARK BY THE EDITOR. 


The fluxion of the algebraic quantity xy is properly yx--ay 
in all cases of increase or decrease. We should always use 
the signs of the fluxions of algebraic expressions as those 
signs arise from the known rules, without considering whether 
the quantities increase or decrease ; but in denoting, algebrai- 
cally, the simple fluxions of geometrical quantities, we should 
prefix the sign minus to the fluxions of such as decrease: and 
thus we may, in any case, use the fluxions of algebraic equa- 
tions, together with the fluxions derived from geometrical 
figures, without embarrasment or apprehension of error. | 


30, We 


316 RULES FOR FINDING 


30, We may now collect all the rules together, which have 
been demonstrated in the foregoing articles, for finding the 
fluxions of all sorts of quantities. And hence, 


Ast, For the fluxion of any Power of a flowing quantity. — 
Multiply all together the exponent of the. power, the fluxion 
of the root, and the power next less by 1 of the same root. 


2d, For the fluxion of the Rectangle of two quantities. —Mul- 
tiply ‘each quantity by the fluxion of the other, and connect 
the two products together by their proper signs. 


3d, For the fluxion of the Continual product of any number 
of flowing quantities. —Multiply the fluxion of each quantity 
by the product of all the other quantities, and connect all the 
products together by their proper signs. 


4th, For the fluxion of a Fraction.—From the fluxion of the 
numerator drawn into the denominator, subtract the fluxion 
of the denominator drawn into the numerator, and divide the 
result by the square of the denominator. 


5th, Or, the 2d, 3d, and 4th cases may be all included under 
one, and performed thus.—Take the fluxion of the given ex- 
| pression as often as there are variable quantities in it, sup- 
posing first only one of them variable, and the rest constant ; 
then another variable, and the rest constant; and so on, till 
they have all in their turns been singly supposed variable, and 
connect all these fluxions together with their own signs. 


6th, For the fluxion of a Logarithm.—Divide the fluxion of 
the quantity by the quantity itself, and multiply the result ny 
the modulus of the system of logarithms. 


Note. The modulus of the hyperbolic logarithms is 1, 
and the modulus of the common logs. is . 043429448. 


7th, For the fluxton of an Exponential quantity having the 
Root Constant. —Multiply altogether, the given quantity the 
fluxion of its exponent, and the hyp. log. of the root. 


8th, For the fluxion of an Exponential quantity having the 
' Root Variable—To the fluxion of the given quantity, found 
by the Ist rule, as if the root only were variable, and the flux- 
ion of the same quantity found by the 7th rule as if the ex- 
ponent only were variable ; and. the sum will be the fluxion 
for both of them variable. 


Note, When the given quantity aonilete of several. terms, 
_ find the fluxion of each term separately, and connect them all 
together with their RB RODRY signs. 


31. PRACTICAL 


© OSH OT bP & De 


oe ee ee er | 
om & Oh = © 


© © © | = = = 
wer oSo aN 


24. 


25. 


%6. 


/ FLUXIONS. 


RULES. 


. The fluxion of axy is 

. The fluxion ef bxyz is 

. The fluxion of cx X (ax —cy) is 
. The fluxion of xy" is 

. The fluxion of x”y"z" is 


The fluxion of («-+y) X(a—y) is 


. The fluxion of 2az? is 
. The fluxion of 223 is 
. The fluxion of 3x4y is 


. The fluxion of Ansys is 


The fluxion of ax2y—x%y? is 
The fluxion of 4a4 —22y-+-3byz is 
t 


. The fluxion of {/x or x” is 
™ 
. The fluxion of (/x™ or x7 Is 


3 1 1 m 
. The fluxion ef —— or _ orxz—7 15 
ym 3 


en 


. The fluxion of 4/ x or x? is 

. The fluxion of 3/ x or n® is 

. The fluxion of 3/2? or x? is 

. The fluxion of ,/z? or x? is 

. The fluxion of 4/x? or at is 

. The fluxion of 3/x‘4 or a? is 

. The fluxion of ,/ (a? + 2?) or (a? +et)Fin is 


The fluxion of ,/ (a? — x*) or (a? 22)? j is 

The fluxion of ,/ (2rz—xa) or a. is 

The fluxion of —— or (ai—x?) 73 
CEs Babi 


The fluxicn of (az—ax)? is 


27. 


3 Os 


_ PRACTICAL EXAMPLES TO EXERCISE THE FOREGOING 


The 


318 


. The fluxion of rae or — 42 
LVAD 


. The fluxion of 


. The fluxion of 


. The fluxion of 


FINDING OF FLUXIONS. 


. The fluxion of 22,/a? £2? is 
. The fluxion of (a? ~x?)? is 


. The fluxion of ,/xz, or (xz)? is 





, eh 
. The fluxion of ,/xz— zz or (az—2z)? is 


$ e 
1s 
axs 


ata 


v2 


is 





. The fluxion of zis 3 
. The fluxion of = is 


e Cc e 
. The fluxion of — is 
oe 


3x 
a—x 





is 


e z Z 
. The fluxion of -—— is 
Lp e 


J ee: 
The fluxion of a is 


. pss 
. The fluxion of —~ is 


y? 


2 
. The fluxion of a is 


~is 


sata 
f(x? —y?) 





. The fluxion of the hyp. log. of az is 

. The fluxion of the hyp. log. of 1 + a is . 

. The fluxion of the hyp. log. of 1 — xis 

. The fluxion of the hyp. log. of x? is 

. The fluxion of the hyp. log. of 4/ zis 

. The fluxion of the byp. log. of 2 is 

| 48. 


The 


FINDING OF FLUENTS. 319 


se Hs Q out 
48. The fluxion of the hyp. log. of zis 


. I-22. 
49. The fluxion of the hyp. log. of = is 
i 1—z 


i+=sz 








50. The fluxion of the hyp. log. of 


51. The fluxion of c* is 

52. The fluxien of 107is 

53. The fluxion of (a+c)* is 

54. The fluxion of 100°Y is 

55. The fluxion of 27 is 

56. The fluxion of y!?* is 

57. The fluxion of 27 is 

58. The fluxion of. (xy)** is 

59. The fluxion of xy is 

60. The fluxion of xy? is 

61. The second fluxion of zy is 

62. The second fluxion of ry, when x is constant, is 
63. The second fluxion of x” is 

64. The third fluxion of x", when x is constant, is © 
65. The third fluxion of xy is 


is 


ype 


THE INVERSE METHOD, OR THE FINDING OF 
FLUENTS. 


32. Ir has been observed, that a Fluent, or Flowing Quan- 
tity, is the variable quantity which is considered as increasing 
or decreasing. Or, the fluent of a given fluxion, is such a 
quantity, that its fluxion, found according to the foregoing rules, 
shall be the same as the fluxion given or proposed. 


33. It may further beobserved, that Contemporary Fluents, 
or Contemporary Fluxions, are such as flow together, or for 
the same time.—When contemporary fluents are always 
equal, or in any constant ratio: then also are their fluxions 
respectively either equal, or in that same constant ratio. 
That is, if e=y, thenisz=y; orif «:y::in: 1, thenis 
Htyi:n:1; orif x =ny, thenis r=ny. 


34. It is easy to find the fluxions to all the given forms of 
fluents; but. on the contrary, it is difficult to find the fluents 
of many given fluxions; and indeed there are numberless 

cases 


320 FINDING OF: FLUENTS., 


cases in which -this cannot at all be done, excepting by the 

_ quadrature and rectification of curve lines, or by logarithms, 
or by infinite series. For, itis only in certain particular forms 
and cases that the fluents of given fluxions can be found; 
there being no method of performing this universally, a priort, 
by a direct investigation, like finding the fluxion of a given 
fluent quantity. Wecan only therefore lay down a few rules 
for such forms of fluxions as we know, from the direct method, 
belong to such and such kinds of flowing: quantities : and 
these rules. it is evident, must chiefly consist in performing 
such operations as are the reverse of those by which the flux- 
ions are found of given fluent quantities. The principal 
cases of which are as follow. | 


35. To find the Fluent of a Simple Fluxion ; or of that in which 
there is no variable quantity, and only one fluxional quantity. 


This is done by barely substituting the variable or flowing 
quantity instead of its fluxion ; being the result or reverse of 
the notation only.—T’hus, 


The fluent of ax is ax. 
The fluent of ay-+ 2y is ay + + 2y. 


The fluent of ./a?+-2? is ,/a?-++2?- 





36. When ny Power of a flowing quantity 2s vein Ves by the 
Elusion of the Root ; 


Then, having substituted, as before, the flowing quantity 
for its fluxion, divide the result by the new index of the pow- 
er. Or, which is the same thing, take out or divide by the 
fluxion of the root ; add 1 to the index of the power ; and 
divide by the index so increased. Which is the reverse of 
the Ist rule for finding fluxions. 


So, if the fluxion proposed be - - 3a5x 
Leave out, or divide by, x then it is - 3x5 5 
add 1 to the index, and itis | ° - 328 ; 
divide by the index 6, and it is Rik ar gue or 425; 


which is the fluent of the proposed fluxion 315 x 
In like manner, ; 


The fluent of 2axx is ax. 
The fluent of 322z is x. ieee 
The 


FINDING OF FLUENTS, gg) 


The fluent of sete is 8 
The fluent of: Qy2 y is 3y 
The fluent of az? z 1s ie 
The fluent of 27 z+ 3y°y y is pabpay?. 
The fluent of z°—}z is ta". 

The fluent of ny"—1y is 


2 
12 
Be 
4 


on 


3 
3° 


The fluent of = or 2722 is 
The fluent of i. is 


The fluent of (a +- x)4z is 
The fluent of (a*-+y*)y3y is 
The fluent of (a3+z3)4z2z is 
The fluent of (a®-+-2")™x"—1; is 
The fluent of (a? ne ee ig 
The fluent of —— 5 Ve ie ae 


The fluent Of ee 
Ke =a) 


37. When the Root under « Vinssihim as a Compound Quantity ; F 
and the Index of the part or factor Without the Vinculum, in- 
creased by 1, ts some aaylane of that under the Vinculum : 


Put a single variable letter for the compound root ; and sub- 
stitute its powers and fluxion instead of those of the same > 
value, in the given quantity ; so will it be reduced toa simpler 
form, to which the preceding rule can then be , applied. 


Thus, if the given fluxion be r= (a?+- x?) 30%, where 3, 
the index of the quantity without the vinculum, increased by 
1, making 4, which is just the double of 2, the exponent of 
x? within the vinculum ; therefore, putting z= a? --2?, thence 
“2 =z — of spe fnxion of which ts 27z =z; hence then x3 
e=ietz = 12 (2 — a?), and the sive al F, or (a? + 


x2) 3¢35 is. = 123; (z—a?), or S1osz —lat2°:; and hence 
the fluent F is = 3.23 — 2 a2z* base8(a2 — 58.07). Or. by 
subetints the value of z instead of it, the same fluent is 3 


(a? x2) 8 x gut - 3,07), ye 3-(a? +2? ) x (2? am 3a?). 
Vou. Il. In 


322 FINDING OF FLUENTS.” 
In like manner for the following examples. _ 
Lo find the fluent of fakin Xx3a, 

To find the fluent of (atber)ta%6, 

To find the fluent of Si ys Xda? zn 

To find the fluent of —— or (or) ae 
; ee aid 





To find the fluent of — or (a-+2"); 2 exter, 
ae 





To find the fluent of ‘verte or (a?-+2? iz 62, 








3 onl 


To find the fluent of 2 Mao ~ oF (a— anit 
fe ~~ 
38. When there are several Terms, involving Two or more Va- 
riable Quantities, having the Fluxion of each multiplied i 
the other Quantity or Quantities : : 


Take the fluent of each term, as if there were sty one 
variable quantity in it, namely, that whose fluxien is contain- 
ed in it, supposing all ‘the others to be constant in that term ; ; 
thenif the fluents of all the terms, so found, be the very same 
quantity in all of them, that quantity will” be the fluent of the 
whole. Which is the reverse of the 5th rule for finding flux- 
jons : Thas, if the given fluxion be xy + zy, then the fluent 
of xy is xy, supposing y constant: and the fluent of ay is also 
sy, supposing x constant : therefore zy is the required fluent 
of the given fluxion aytay. | | 


Tn like manner, 

‘The fluent of xyz--ay z+ ay2 is ee; : 
‘The fluent of Zaye tat y is aye Renee a 
| The fluent of ly? oy HHP yy i is ; 
The fluent of I= op Face Aa 

ye ye 





4 Pare e aij 
2 —lax2y-7y 2 ax? 
The fluent of ane chic Ne weak a A or ne a 


- is 
SY BS9 


a9, W. tok 


FINDING OF FLUENTS. 398: 


39. When the. given ‘Fluxional Expression 2s in this Form 22-29 any 


namely, a Fraction, including Tv Quantities, being the F lux- 
ion of the Former of them drawn into the latter, minus the 
Fluxion of the latter drawn into the former, and divided by 
the Square of ine latter : 


Then, the fluent is the fraction — —, or the tore ny 
divided by the latter. That is, 


The fluent of —/——2 a Be a rs ive tanning, 
— Pe2 a>; 5] 
The fluent of Sexy — ft Wy is ae 


Though, indeed, the examples of this case may be perform- 
ed by the foregoing one. Thus, the given fluxion - -  - 


ds eet | reduces to Tae or ytd y~?; of which, 


the fluent of 2 3 ite supposing y constant; and 
the fluent of —2xyy-? is also 8. ore, when x is constant : 


therefore, by that case, — - * is the fluent of the whale <. 


40. When the F luxion of & Quantity is Divided by. the Suan 
ty ttself:: 


‘Then the Hick is equal to the hyperbolic fosarithis of that 
quantity ; or, which is the same thing, the fluent is equal to 
-2°30258509 multiplied by tae common logarithm a Bees same 
quantity. 


| So, the fluent of 2 or 2-14, is the hyp. log. of Bis ci be 
The fluent of = is 2 X hyp. log. of AG or = hyp. log 2. 


The fluent of is a Xhyp. log. 2, or mares ee of x2. 





The dues N ged is 


= Ee 





41. Many 


324 FINDING OF FLUENTS. 


41. Many fluents may be found by the Direct Method thus : 
Take the fluxion again of the given fluxion, or the second 


| ak 
fluxion of the fluent sought; into which substitute se. for x 


Bo fay s y, &c.; that is, make z, z, x, as also y, y, y, &c. to 


be in continual proportion, or so that: «:: 2: 2, and - - 
Yy: sy: ys &c.; then divide the square of the given flux- 
ional expression by the second fluxion, just found, and the quo- 
tient will be the fluent required in many cases. 


Or the same rule may be otherwise delivered thus: 


In the given fluxion fr, write x for x, y for y, &c., and call 
the result a, taking also the fluxion of this quantity ¢ c ; then 
make G : ¥ :: G:F; so shall the fourth proportional r be the 
fluent sought i in many cases. 

It may be proved if this be the true fluent, by taking the 
fluxion of it again, which if it agree with the proposed fluxion, 
will show that the fluent is right; otherwise, it is wrong. 


EXAMPLE. 


Exam. 1. Letit be required to find the fluent of az"—1z. 


Here r==nz"—!;. Write x for x, then nz°—'x or na" == ; 
the fluxion of this is ¢ =n? itn therefore G:F ::6:F, 
becomes n?a°—!z : na®—lzy :: na: gt == PF, the fluent sought. 


Exam. 9. To find the fluent of 3 y-+-ay. 


Here r= 2y+sry; then, writing x for « and y for y, it is 
xy+xy or 2Zry=c ; hence eo Ray tory 5 theng :F::G6:F, 
becomes Ory Qny : xytay :: 2xy . ay==F, the fluent sought. 


_ 42. To find Fluents by means of a Table of Forms of Fluxions 
, ' and Fluents. 


In the following Table are contained the most usual forms 
of fluxions that occur in the practical solution of problems, 
with their corresponding fluents set opposite to them; by 
means of which, namely, by comparing any proposed fluxion 
with the corresponding form in the table, the fluent of it will 
be found. — 


Forms. 


FORMS OF FLUXIONS AND FLUENTS. 2 825 


Forms. Fluxvons. Fluents. 


REE Cece ee 











ee 































































































































































































Oe ce Wat 
7 
IJ (ataryn—ign's -—— (azbat)? 
———- — en 
gmn—1), : 1 ; gma pi 
a Chie gt) tip Es “Tanna * (ata 
(aba®)>—1,; ap ae — 1. (atan)m 
IV gmat? jmna x gin | | 
(myz epney) Peer ry : 
V , emy® 
or <= ere 
maem—l' yr2" nary ly 24ra™ iat le: : 
vi (°° (may2+ ee rxyz jas year}, yet 
or (SE pL LTE) amyte 
mod , 
VII FE ora-tz | , log. of x. 
a ee is 5) eae 
Vir log. of at x 
a+ 2p 
xaly | n | 
— log. of— 
she a+Xy, oO  a@tkan 
7 
n—1 a: n 
x {ee a ag leh 
aan VA f &— Vf x 
put: hile op tae tan ¢/-; or 
x1 Pee | n/a os 
a-Pae 1 She 
——- X are to cosine 
NJQ | Ly RE 
1 
XII TST aes 2, : ee so 
State “log..0 of 4 + ftata' 


A LOI I LD IP GIRS Ee eae cet nD TON 


Forms. 


326 FORMS OF FLUXIONS AND FLUENTS. 



































































































































(Forms.|  Fluxions. * Fluents. B 
ae 2 ; gn $i) 
| He ae ~ X arc to sin. 4/—, or : 
rs 1. Qy2 
| Asli ~ X arc to vers. —— 
n ‘a 
an 1 a ea ee gt 
SLY uo —— log. of Vat SR 
fia ee ei Jrtet sa 
x arc to a x or 
xv |= | nsf Oo. : Gg? 
Rios ee 1 -  2a—2" 
fare X are to cosin. peu 
Naif & 
———— Ae : Ey 
XVI |2./fde —x7 5 circ. seg. to diam. d & vers. x. 
nz . 
XVI lex teas. te 
_|n log. c 








eR 








ed 





XVI |oys log. yay ly 








os 


Note. The. logarithms, in the above forms, are the hyper- 
bolic ones, which are found by multiplying the common loga- 
rithms: by 2302585092994. And the arcs, whose sine, or 
tangent, &c. are mentioned, have the radius 1, and are those 
in the common tables of sines, tangents, and secants. Also, 
the numbers m,n, &c. and to be. some real quantities, as the 
forms fail when m =0, or n=O, &c. 


The Use of the foregoing Table of Forms of Fluxions and Fluents. 


43. In using the foregoing table, it is to be observed, that 
the first column serves only to show the number of the forms ; 
in the second column are the several forms of fluxions, which 
are of different kinds or classes ; and in the third or last co- 
lumn, are the corresponding fluents. 
The method of using the table, is this. Having any 
fluxion given, to find its fluent: First, Compare the given 
- fluxion with the several forms of fluxions in the second co- 
lumn of the table, till one of the forms be found that agrees 
‘with it; which is done by comparing the terms of the given 
fluxion with the like parts of the tabular fluxion, namely, 
the radical quantity of the one, with that of the other ; and 
the 


FINDING OF FLUENTS. 327 


the exponents of the variable quantities of each, both within 
and without the vinculum ; all which, bemg found to agree or 
correspond, will give the particular values of the general 
quantities in the tabular form : then substitute these particu- 
Jar values in the general or tabular form of the fluent, and the 
result will be the particular fluent of the given fluxion: after 
it is multiplied by any coefficient the proposed fluxion may 
have. oe | 


EXAMPLES. 


Exam. 1. To find the fluent of the fluxion S025, 


This is found to agree with the first form. And, by com- 
paring the fluxions, it appears that x = 2, andn — 1 = §, or 
n==% ; which being substituted in the tabular fluent, or Dip, 
gives, after multiplying by 3, the coefficient, 3X203, or a5, 
for the fluent sought. | : 


Exam. 2. To find the fluent of 5a? /c3- —«3,0r 5x24 (coma?)?, 


This fluxion, it appears, belongs to the 2d. tabular form ; 
for a= c*, and — «°= — x*, and n= 3 under the vinculum, 
also m—1==1, or m=, and the exponent ®—? of x°—! with- 
out the vinculum, by using 3 for n,is 7 — 1=2, which agrees 
with x? in the given fluxion: so that all the parts of the form 
are found to correspond. Then, substituting these values into 


the general fluent, _— (a—x")™, ; 
; 3 etry 3 
it becomes —£ X2 (c3 ~%3)? =— 1 (c3—a3)?, 
x2» 
Exam. 3. To find the fluent of euits 
This is found to agree with the 8th form; where - - -— 
toe==--x? in the denominator, orn=3 ; and the numerator 
x*—1 then becomes x?, which agrees with the numerator. in 
the given fluxion ; alsoa—=1. Hence then, by substituting in 
the general or tabular fluent, 5 log. of a + 2", it becomes } 
log. 1-++-23. ! ' 
Exam. 4. To find the fluent of ax*z. 
Exam. 5, To find the fluent of 2 (10-2? yea. 
. no ates ae 
Exam, 6. To find the fluent of (ebay 


Exam. 7. To find the fluent of See 
(a—x)* Exam, 8. 





328 


FINDING OF FLUENTS. 


com 2. 


. 8. To find the fluent. of © cg 


. 9. To find the Aliet of 


+ 29s 


26, 


. To find the fluent of 5 ie 

. To find the fluent of 3x 
. oe 

. To find the fluent of ee 

£—= 3x3 


. To-find the fluent of as 


3 
3 
. To find the fluent of 5 raed 


. To find ne Avent Ob 


. To find the fluent of 
. To find the fluent of 
. To find the fluent of 
. To find the fluent of —— 


. To find the fluent hee 


14+, 





. To find the fluent of 4% dy? 
BN gt 
. To find the fluent of fees 3y) xy". 


3x 3 q 
. To find the fluent of — or — a—1}. 
ax’ $a 





— Ir 








— x? D inne, 














en 4” 
2 
any 


aie 








a My 














sei 








jae 


arte 








fi a 


To find the fluent of ——__——., 
ee as 





To find the fluent of’ 





fal 


Exam. 27. 


‘A 


FINDING OF FLUENTS. 


03 
to 
Oo 





Exam. 27. To find the fluent of say 
of ryan? 
Exam. 28. To find the fluent of 2% / 2a—2?. 
Exam. 29. To find the fluent of arx. 
Exam. 30. To find the fluent of 3a?7x. 
Exam. 31. To find the fluent of 3z7z log. z-+- 3x2z*-!z. 


Exam. 32. To find the fluent of (1+ 23) zz. 


Exam. 33. To find the fluent of (2 + #4) a?ic 
Exam. 34. To find the fluent of 22x + a? +23. 


To find Fluents by Infinite Series. 


44. When a given fluxion, whose fluent is required, is so 
complex, that it cannot be made to agree with any of the 
forms of the foregoing table of cases, nor made out from the 
general rules before given ; recourse may then be had to the 
method of infinite series ; which is thus performed : 


Expand the radical or fraction, in the given fluxion, into 
an infinite series of simple terms, by the methods given for 
that purpose in books of algebra; viz. either by division or 
extraction of roots, or by the binomial theorem, &c.; and 
multiply every term by the fluxional letter, and by such sim- 
ple variable factor as the given fluxional expression may con- 
tain. Then take the fluent of each term separately, by the 
foregoing rules, connecting them altogether by their proper 
signs ; and the series will be the fluent sought, after it is mul- 
tiplied by any constant factor or coefficient which may be 
contained in the given fluxional expression. 


45. It is to be neted however, that the quantities must be 
so arranged, as that the series produced may be a converging | 
one, rather than diverging : and this is effected by placing the 
greater terms foremost in the given fluxion. When these are 
known or constant quantities, the infinite series will be an 
ascending one ; that is, the powers of the variable quantity 
will ascend or increase; but if the variable quantity be set 
foremost, the infinite series produced will be a descending one, 
or the powers of that quantity will decrease always more and 
more in the succeeding terms, or increase in the denominators 
of them, which is the same thing. 

Vor. TF. #3 For 


330 FINDING OF FLUENTS. 


For example, to find the fluent of --— ; en 

Here, by dividing the numerator by the gee the 
proposed fluxion becomes x — 2az-+-32? x — 5x3 x+824x— Xe; 
then the fluents of all the terms being taken, give - - - 

Ge? --x? —Sx4-+345 —&c. for the fluent sought. 

Again, to find the fluent of 3+ 1—2?. 

rie; by extracting the root, or expanding the radical 
quantity «/ 1~23, the given fluxion becomes - - 
x—Z0 c—teig—hx'x—&c. Then the fluents bf all the 
_ terms, being taken, give c—12°— 525 —;1,27 ~&c. for the 
fluent sought. 


OTHER EXAMPLES. 


Exam. 1. To find the fluent of rar both in an ascending and 


Q sme Y 





descending series. 
Exam. 2. To find the fluent of et in both series. 
(a— ae 
1—<z? 2-204 ‘ 
I-pa--% a—g2 
to find z. 





Exam. 3. To find the fluent of 


Exam. 4. To find the fluent of 





Exam. 5. Given 2 i ay eras es 
3 2 
ee to find z. 
Exam. 7. Given z= 3x a--2, to find z. 
Exam. 8. Given == = Qn fa?+ +22, to find z. 


Exam. 9, Given 2= 4x a? —2?, to findz. 





Exam. 6. Given == 














er Sax 
Exam. 10. Givenz = i) —., to find z. 
af x? = (2 
Exam. 11. Given z = 2x3/a?—z°, to find z. 
dixam. 12. Given z= - Saat Ate to find z. 
MOREE ; 


Exam. 13. Given >= = 2n9/n?+et+et, to find z. 
Exam. 14, Given z= 52 +f ax —az, to find z. 


To 


FINDING OF FLUENTS. 33) 


To Correct the Fluent of any Given Flucion. 


46. The fluxion found froma given fluent, i is always per- 
fect and complete ; but the fluent found froma given fluxion 
is not always so; as it often wants. a correction, to make it 
contemporaneous with that required by the problem under 
consideration, &c.: for, the fluent of any given fluxion, as z 
may be either xz, which is found by the rule, or it may be a-¢c, 
or x ~c, that is 2 plus or minus some constant quantity c; be- 
cause both x and tc have the same fluxion « x, and the finding 
of the constant quantity ¢, to be added or subtracted with the 
fluent as found by the foregoing rules, is called correcting the 
fluent. 


Now this correction is to be determined from the nature of 
the problemin hand, by which we come to know the relation 
which the fluent quantities have to each other at some’ certain 
point ortime. Reduce, therefore, the general fluential equa- 
tion, supposed to be found by the foregoing rules, to that point 
or time ; then if the equation be true, it is correct ; but if not, 
it wants acorrection; and the quantity of the correction, is 
the difference between the two general sides of the equation 
when reduced to that particular point. Hence the general 
rule for the correction is this : 


Connect the constant, but indeterminate, quantity c, with 
one side of the fluential equation, as determined by the fore- 
going rules ; then, in this equation, substitute for the variable 
quantities, such values as they are known to have at any par- 
ticular state, place, or time ; and then, from that particular 
state of the equation, find the value of c, the constant quan- 
tity of the correction. 


EXAMPLES. 


47. Exam. 1. To find the correct fluent of z—azr?z. 


The general fluent is z=ax4, or z==-ax' + c, taking in the 
correction c. 


Now, if it be known that z and x begin together, or that z 

_ is =0, when x —0; then writing O for both x and z, the ge- 
neral equation becomes 0—=0--c, or =c; so that, the value 

of c being 0, the correct fluents are z=az?. we 
u 


332 FINDING OF FLUENTS. 


But if z be =0, when « is =}, any known quantity; then 
substituting 0 for z, and 6 for x, in the general equation, it be- 
comes 0 ab4-+c, and hence we find c= —ab* ; which being 
written for c in the general fluential equation, it becomes z= 
az’ —ab‘, for the correct fluents. 


- Or, if it be known that 2 is = some quantity d, when x is 
==some other quantity as 6; then substituting d for z, and 6 
for x, in the general fluential equation z—=ax4-+-c, it becomes 
d==ab4-}-c; and hence is deduced the value of the correction, ~ 
namely, c==d—ab4 ; consequently, writing this value for c in 
‘the general equation, it becomes z= axr4 — ab+ + d, for the 
correct equation of the fluents in this case. 


48. And hence arises another easy and general way of cor- 
recting the fluents, which is this: In the general equation 
of the fluents write the particular values of the quantities 
which they are known to have at any certain time or position ; 
then subtract the sides of the resulting particular equation — 
from the corresponding sides of the general one, and the re- 
mainders will give the correct equation of the fluents sought. 


So, the general equation being z= ax! ; 

write d for z, and 6 for x, then d=ab* ; 

hence, by subtraction, - z2—d=—azx’ —ab‘ ; 

or z=az4—ab4--d, the correct fluents as before. 


Exam. 2. To find the correct fluents of 2= 5xz s° 2 being 
~ ==0 when z is =a. 


Exam. 3. To find the correct fluents of z=Sz 4/a-+< ; 
z and x being =0 at the same time. 


psi eup 4? apuaane 
Exam. 4. To find the correct fluent of = ; supposing 
x 





zand 2 to begin to flow together, or to be each = 0 at the 
same time. 


| | by ; 
Exam. 5. To find the correct fluents of Joakim suppos- 


“Ing z and x to begin together. 


Ant. 49. 


[333 J 
OF FLUXIONS AND FLUENTS. 


Arr. 49. In art 42, &c. is given a compendious table of 
various forms of fluxions and fluents, the truth of which it may 
be proper here in the first place to prove. . | 

50. As to most of those forms indeed, they will be easily 
proved, by only taking the fluxions of the forms of fluents, 
in the last column, by means of the rules before given in 
art, 30 of the direct method ; by which they will be found to 
produce the corresponding fluxions in the 2d column of the 
table. . Thus, the ist and 2d forms of fluents will be proved 
by the Ist of the said rules for fluxions ; the 3d and 4th forms, 
of fluents by the 4th rule for fluxions ; the 5th and 6th forms, 
_ by the 3d rule of fluxions ; the 7th, 8th, 9th, 10th, 12th, 14th 
forms, by the 6th rule of fluxions: the 17th form, by the 7th 
Tule of fluxions : the 18th form, by the 8th rule of fluxions. 
So that there remains only to prove the 11th, 13th, 15th, and 
16th forms. | 

51. Now, as to the 16th form, that is proved by the 2d ex- 
ample in art. 98, where it appears that 2,/ (dx—a?) is the 
fluxion of the circular segment, whose diameter is d, and 
versed sinex. And the remaining three forms, viz. the 11th, 
13th, and 15th, will be proved by means of the rectifications 
of circular arcs, im art. 96. : 

52. Thus, for the 11th form, it appears by that art. that the 
fluxion of the circular arc z, whose radius is r and tangent ¢, 











m 
° ° Als rag | nr Vie 
82 Sey Now put ¢ = 22", or#??= 2 ,-and a = 7? : 
r2 : 
nah In—}. n : r2¢ 
thenis¢=inz2 x, andr? + 22 =a-+2z,and z= . 
% j 72 {4° 
in—t1, in—1, : 9 
Lane? >> an. rs a va : F 
== ———~ 5 liénce Sao 2, and the~ fluent is. 
Apa | a2" lon an ; 
oz 


2 : in Q . 
—=— X arc to radius ,/ a and tang. x2 or=——X arc to 
an na Af 


radius 1 and tang. oe which is the first form of the fluent 
in n°. xr. ee 
53. And, for the latter form of the fluent in the same n°; — 


lg 
is dou- 


2 


because the coefficient of the former of these, viz. 
1 


N/a 
in. the latter case, must be double the arc in the former 
But the cosine of double an arc, to radins 1 and tangent f, 


#2 





ble of 





the coefficient of the latter, therefore the arc 


334 FLUXIONS AND FLUENTS. 


of -and because fa ty the former case, this substi- 


see 


tuted for #2 in the éaehelin 





"it becomes — ———; ~ the cosine 


au wr 


as in the latter case of the 11th form. 


54, Again, for the first case of the fluent in the 13th 
form. By art. 61, ae Adston of the circular arc 2, to radius r 


aio 
and sine y, 1s need aay to the radius I. 
Jr =¥y ° (1-9?) 


Now mindgerat) hence %, (1 — ¥?) a 


in—1. 


V-S)=y- X of (a—z"), and y= +f — ra nat x3 


then these two ‘Dene substituted in the value of z, give z 
aot 
n x? 


Couey : | 
or Asia By: ae eee fa on : coneqeéntly the given fluxion 


2. i ne La let. 
=o 2 and therefore its fluent is 72 that 1s = x 


_are to sine 4/— Gas in the table of forms, for the first case of 


form x1II. 


| seeps : . 
55, And, as the coefficient = in the latter case of the said 


form, is the half of = the coefficient in the former case, there- 


fore the arc in the latter case must be double of the arc in 
the former. But, by trigonometry, the versed sine of double 
an arc, to sine y and radius 1, is 2y? ; and, by the former case 


n 7 


2 1 22 
Qy2 = — 5 therefore : X arc to the versed sine en is the 


fluent, as in the 2d case of form xi1r. 
56. Again, for the first case of fluent in the 15th form. 
By art 61, the fluxion of the circular arc z, to radius 7 
r2 ¢ 
sy (SF a93)?° acne —1) 


Now, put s = vf pear 


and secant s, is z= to radius 1 





, or s? == ; hence sy (s?—1) = 
in an 1 io 

i i is rds a is ' P 

Var a ree and s = v- X ing 9x7 

| then 


: 


'FLUXIONS AND FLUENTS, 335 


then these two being substituted in the value of 2, give = or 


$ n/a xx - . 
———_ ++. —_—_——_; consequently the given fluxio1 
sV(s?—1) 2 '4/(x"—a)’ sp all : 
alg 





— 
— 


V(a—a) nfo 
X arc to secant ,/ —, as in the table of forms, for the first 





° 4 e 2g e 
z, and theref. its fluent is ——— z, that is 
nfo 


case of form xv. 
57. And, as the coefficient 


2 
nf o 
case, therefore the arc in the latter case must be double the 
arc in the former. But, by trigonometry, the cosine of the 





;in the latter case of the 
naf a 


said form, is the half of the coefficient of the former 





double arc to secant s and radius 1, is mee 1; and, by the 








2 2a 2a—a" 1 
former case, —-—1=—_ —1= —; therefore ——% arc 
s? Ln tn na a 
~ ,  2a—an. apy 
to cosine —-—— is the fluent, as in the 2d case of form xv. 


: 2 M a 
Or, the same fluent will be Ae X arc to cosine ,/ —~) be- 


cause the cosine of an arc, is the reciprocal of its secant. 
58. It has been just above remarked, that. several of the 
tabular forms of fluents are easily shown to be true, by taking 
the fluxions of those forms, and finding they come out the 
same as the given fluxions. Butthey may also be determined 
in a more direct manner, by the transformation of the given 
fluxions to another form. Thus, omitting the first form, as 
too evident to need any explanation, the 2d form is z= (a +- 
Ln)™—14"—1z, where the exponent (nm — 1) of the unknown 
quantity without the vinculum, is 1 less than (n), that under 
the same. Here, putting y= the compound quantity a-+-2" : 


ym—ly 
nm 





; Tr 
then is y =nx"—‘z, and z= ; hence by art. 36, z = 


n\m T 
=e fora") as in the table. 
mn 


59. By the above example it appears that such form of 
fluxion admits of a fluent in finite terms, when the index (n— 1) 
of the variable quantity (x) without the vinculum, is less by 
1 than n, the index of the same quantity under the vinculum. 
But it will also be found, by a like process, that the same’ 
thing takes place in such forms as (a-+-x7)™x°"—1z, where the 

exponent 


336 FLUXIONS AND FLUENTS. 


exponent (en — 1) without the vinculum, is 1 less’ than any 


- multiple (c) of that (n) under the vinculum. And further, 


> 


2 


ney = y, OF 21g = 


that the fluent, in each case, will consist of as many terms as 
are denoted by the integer number c; viz. of one term when 
e== 1, of two terms when c = 2, of three terms when c= 38, 
and so on. 

60. ‘Thus, in the general form, z= (a-+-x")"«"—1z, put- 
ting as before, a--a" =Y 5 then is 2” = y — a, and its fluxion 
wr and gor—ly or go" "1, es 1 

b 
ne n 
( (ya) "7; also (a--x7)"= 7" : these values being now sub- 


stituted in the general form proposed, give p= i(y= =a)" Ye 


Now, if the compound quantity (y—a)*—' be expanded by the 
binomial theorem, and each term multiplyed by yy, that flux- 
ion becomes 

pate Lo mte—1; “pet shee? C— 1. ern% 
&c.); then the fluent of every term, being taken by art. 36, itis 
ee 1 ante Cu guyete—t ec—le—2 arymbome 





a2 a pea ui 




















= -(——- tH —'&e.), 
p ne 1 m+c— 1 1 2 m+c—2 Be 

a c—la c—l.c=2 a? seer dew aires) a3 
ae Gr d= fy). d— 235242 d—3 " 9.3y3 


&c.), putting d= m +c, for the general form of the fluent ; 
where, c being a whole nausber the multipliers c—1, cua. 
c—3, &c. will become equal to nothing, after the first c ‘terms, 
and therefore the series will then terminate, and exhibit the 
fluent in that number of terms; viz. there will be only the 
first term when c = 1, but the first two terms when c = 2, 
and the first three terms when c= 3, and so on. —Except 
however the cases in which m is some negative number equal 
to or less than c; in which case the: divisore: m--c, m+c—1, 
m-+¢c — 2, &c. becoming equal to nothing, before the multi- 
pliers c—1, c—2, &c. the corresponding terms of the series, 
being divided by 0, will be infinite; and then the fluent is 
my ” fail, as in such case nothing can be determined from it. 
- Besides this form of the fluent, there are other. methods 
of Bonne by which other forma of fluents are derived, 
of the given fluxion z= (a + 2”)™x"—'z, which are of use 
when the foregoing form fails, or runs into an infinite series ;. 
some results of which are given both by Mr. Simpson and 


Mr. Landen. The two following processes are after the man- 


ner of the former author. 
62. The given fluxion being (a--a")™z°"—)z ; its fluent may 
: me 


FLUXIONS AND FLUENTS. 337 


be assumed equal to (a + «”)”*! multiplied by a general se- 
ries, in terms of the powers of z combined with assumed un- 
known ceefficients, which series may be either ascending or 
descending, that 1s, having the indices either increasing or de- 
creasing :. Fi ee 

viz. (aba )"*! X (ax Be’ --cx"~*-- pa? &c.), 

or (a-f-a”)™*1 X (aa Ba’* cats pg + &e.). 

And first, for the former of these, take its fluxion in the usual 
way, which put equal to the given fluxion (a-a")"a-"—"1z7,, 
then divide the whole equation by the factors that may be 
common to all the terms; after which, by comparing the like 
“indices and the coefficients of the like terms, the values of 
the assumed indices and coefficients will be determined, and 
consequently the whole fluent. Thus, the former assumed 
Series in fluxions is, — : 
2 {m+ 1) ae (a 2")™ XK (ant + Bar + cx?—* &e.) + 
(a+ a") "2X (rax?—! -- (r—s) Ba! + (r — Qs)ex7—2-! 
&c.); this being put equal to the given fluxion (a--2")"x"—1y, 
and the whole equation divided by (a+")”a—!z there results 
n (m+ 1) a" x (AnT-- Ba’? + cx? —* 4- pas 4 &e,) 
++ (a + 2”) x (rax7-+-(r—s)Ba"—-+-(r — 28)cx7—**&c. ) ¢ ea 
Hence, by actually multiplying, and collecting the coefficients 
of the like powers of x, there results 
z(m+-1) aart".in(m-+1) satchel carn, 
rn +-—rs -+-r—2s 
—20".. fe... roan”... - (r—s) apr’ &ec. 
Here, by comparing the greatestiindices of x, in the first and 
second terms, it gives r + » = cn, and r+n—s=r; 
which give r == (c—1)n, anda =s. Then these values be- 
ing substituted jy, the last series, it becomes 
(c+m)nax"+(c+-m—1)nex"—"-+-(c-+-m— 2)nce"—*"&e. an 
— x" 4-(¢ ~ 1)naax—"-+4-(c—2)napa—" &e, as 
Now, comparing the coefficients of the like terms, and putting 
c-++-m==d, there results these equalities : 

Neos So l.aa ¢—1l.a- c—2.aB__ , ¢=1.c-2.a? 
dy dod. hadith. dss, Seti eid’ 
&c.; which values of a, 8, c, &c. with those of r and s, being 
now substituted in the first assumed fluent, it becomes 
(afarymtlyon—n 1 ela . e—lic—2.a2 c—1.c—2.c—3.03 

ii 1 doa d-1.-2.09* d-1d8 3.0 

5 &c. the true fluent of (a + 2")™2°"—1z, exactly agreeing 

with the first value of the 19th form in the table of fluents 

in my Dictionary.” Which fluent therefore, when c is a whole 

- positive number, will always terminate in that number of 
terms ; subject to the same exception as in the former case. 

Vor. HH. | 44 Thus, 


=0. 








338 FLUXIONS AND FLUENTS. 


Thus, if ¢c =2, or the given fluxion be (a + a”)"4*—1, ; 
then, c+ m or d being = m + 2 the fluent becomes 
(aban) aca 1 faa (m+ 1)an—a 














(m--2)n ( paeerar th n m+1 m2 
And if c = 3, or the given fluxion be (a--an)*x3"—lx ; 
then m -- c or d being = m + 3, the fluent becomes 
 (a-pa”)nt lean Pax” Vary 2” ch Gea tal “yet 
“(m-+3)n m+? m+2m+1? n m+ 3 
sda Sa hs aha ———), And so on, when ¢ is other. 
m-+-3.m+4+2  m+-3. m+2.m-+1 : 


whole numbers: but, when c denotes either a fraction ora 
negative number, the series will then be an infinite one, as 
none of the multipliers o — 1,c—2,c—3, can then be equal 
to nothing. 

63. Again, for the latter or ascending form, (a-Panymtt 
Cag -b Bat + ca7t2s + pats + &c.), by making its fluxion 
equal to the. proposed one, and dividing, &c. as before, equat- 
ing the two least indices, be, the: fluent will be obtained in a 
different form, which will be useful in many cases, when the 
foregoing one ‘fails, or runs into an infinite series. Thus, if 
r-+s,r-+ 2s, &c. be written instead of r— s, 7 — 2s, &c. 
respectively, in the general equation in the last case, and 
taking the first term of the 2d line into the first line, there 
results 


Be ies ces Cau ere ee ae 
seraace (rt s)ana"*4(r 2s)aex”* &c. 


Here, comparing the two least pairs of exponents, and the 
1 Lb: 
coefficients, we have r=cn, ands==n; then Aree es 


r--n(m+1)_ emt A ‘e+m-+1 
A == == et ee ee | el SD g 
~ a(r-Fs) ; _eFl ‘a (c+ 1)cna2 ’ 
_ckm+2 c+-m-+-1.e-+-m-+-2. 
ea eee p= pte: ke Therefore, denot 
ing c -- m by d, as before, the fluent of the same fluxion™ 
(a--x")”ae"—1 5, will also be truly. expressed by 
2 Sere Mi 1 d+1.2" , d+1.d+2.22" 
cna Go ‘e--L.. aah e+ 1.c+2.a2 
agreeing with the 2d value of fhe fluent of the 19th form in 
my Dictionary. Which series will terminate when d or e--m 
is a negative integer ; ; except when c is also a negative inte- 
ger less than d; for then the fluent fails, or will be infinite, 
the divisor in that case first becoming equal to nothing. — 


B= — 








eae irs 


To 


FLUXIONS AND FLUENTS. 339... 


To show now the use of the foréeoine series, in some exam- 
ples of finding fluents, take first, 


64, Example 1. To find the fluent of - Sak ts) or 6x 
1 
{a-+2)?. 


This example being compared with the general form 
an—ly (a-+a")”, in the several corresponding parts of the 
first series, gives these following equalities: viz.a=a,n=1, 
cn — fa% or c= ia Si a m= —1; ya, 

Ti 2 om Bes 2a 
—_; —— I ce ae ° 
d==m--c=—=2—1==3, ay" =(a +2)", = Raat y orn 
here the series ends, as all the terms after this become equal 
to nothing, because the following terms contain oe er c— 





@—0. These values then being substituted in — ts d ~ -- = 
— a\t 
- ay ai . 
it becomes (a-+- x)’ X : ~=)= ma 2a) (a 1) 


ee 4a 





V¥ (a +2) ; which multiplied . 6, the given co- 


spicient in the proposed example, there results Se ebae VA 
{a-+-«), for the fluent required. 
17. Exam. 2. To find the fluent 2s 
2 
Sea Con Toth pe = (a? +22)? X3a—5y, 
; oS Soa gee 
The several parts of this quantity being compared with the. 
corresponding ones of: the general form, give a = a?; n= 2, 


1—6 
m=}, (n= 1 or 2cm1= —6, whence c= = —3 and d= 





m-+-c=1—$—= —4= —2, which being a negative, integer, the 
fluent will be obtained by the 3d or last form of series ; which 


S(at-+01)hans 


on substituting these values of the letters, gives — Ba 


2 een 


— ~ gf 





1-1. aye 3(a? +22)? 
J 20? / > barat 
2x3 — 3a? 
5a* 
66. Exam. 3. Let the fluxion proposed be 
ogc that ts Beas 
eee) eee 
Here, by proceeding as before, we have a==b, n=n, m=—t, 
e=3, and d=c-+-m=5 ; wherec being a positive integer, this 
case 








x(i-— 


for the required fluent of the proposed fluxion. 


340 FLUXIONS AND FLUENTS: 
case belongs to the 2d series ; into which therefore the 


b n\2 ay 
above. values being substituted, it becomes —~ af bb) 2! 


gn 
47° 1. 6 Ba2*—4ba"-4-8b? 
G-= —+ 3. 1 Lyn” =2,/(b+2 *) x 3n 


1 
67. ‘ee 4. Let the proposed fluxion be 5 (328 )tente, 
Here, proceeding as above, on have a=1, n=2, m=1, cn 
—lor 2c—1=—8, and c= —1, c= —z, d= c+t+m= —3, 
which being a negative integer, the case belongs to the 3d or 
last series ; slain therefore, by substituting these values bes 




















5(L—22)2 beak ns pel aes 18 —2)3 
comes a ae mena eae b =—727 


3 

2° 94274 _3(1L.— 22)? 
Coa ——)= ==") X(5-+ 1222-2424), the true 
fluent of the proposed fuxion. And thus may many other si- 
milar fluents be exhibited in finite terms, as in these following 


examples for practice. 
Ex. 5. To find the fluent of —3z3z Ja —z?). 


Ez. 6. To find the fluent of —6x5x . (a? nae? 
’ sz 
Ex. 7. To find the flu. of wucisi 78 


OT (a—2")?x 
girl 
68. The case mentioned in art. 37, viz. of compound quan- 
tities under the vinculum, the fluxion of which is in a given 
ratio to the fluxion without the vinculum, with-only one varia- 
ble letter, will equally apply when the compound quantities 
consist of several variables. Thus, 
Example. 1. The given fluxion being (422+ yy) X of 
(x3--2y?), or (412 +8yy) X (2? +22), the root being 2? 
+2y?, the fluxion of which is 2xz +4yy. Dividing the former 


fluxional part by this fluxion, gives the quotient 2: next, the 
exponent 1 increased by 1, gives 3: lastly, dividing by this 


%, there then results 4 (2?-+-2y? i, for the rereret fluent of 
the proposed fluxion. 


nt1>., 


Exam. 2. In like manner the fluent of — 
(22> yt + 4.28)5 X (Gaz + 12y?y -+1825z) is 
af : q 4 
(08 -f-yt | 2° )9* 4X (Crane 1oyey Ph lbztz oe ey. 3 
(Bae +4y?y +6282) X4 aa SMU Se 1 
Exam. 3. In like manner, the fluent of 


n? (ay*--ayy+a%2 ) sf (a2 +2y?), is 2 5 (at-baxtye th 
e@ 


/ 


FLUXIONS AND FLUENTS. 341 


69. The fluents of fluxions of the forms. 
gn ae a” x cn—1} > 
zsba’ a? ba?’ 
numbers, will be found in. finite terms, by dividing the nume- 
rator by the denominator, using the variable letter x as the 
first term of the divisor, continuing the division till the powers 
of x are exhausted ; after-which, the last remainder will be 
the fluxion of a logarithm, or of a circular nc &e: / 


Exam. 1. To find the fluent af 


: xz 
&c. or 


an” Ge a” 





—, &e. where c tnd m are whole 





= | a 
Beis be 5 a Oe . Werk. 
| mae =7 ice" where the remainder prep: is €V1- 
dently =< a Xthe fluxion of the hyperbolic logarithm of a+ : 
therefore the whole fluent of the proposed fluxion is x~a 
hyp. log. of (a+). In mee manner it will be found that, 


Ex. 2. The fluent of <=, is x-+a X hyp. log. of (era). 


Ry division, — 





Ex. 3. The fluent of =. isco aca hyp. log. of (a—x). 


Ex. 4. The fluent fe 
(a-+2). 

Ex. 5. The fluent of — =, is — 327 — 0% == a? X hyp. log. of 
(a—2). ; % 

Ex. 6. The fluent of = 
| (ea). 

Ex. 7. The fluent of = 
log. of (x--a). ! 

Ex: 8. The fluent of =, is 12°-P lax? +a924-a% Xhyp. 





are oe: Bad gan alse hyp. log. of 





mare =, is ya" at + a? X hyp. '0s- of 





im is 1¢3— jos Base Las tv's. 


log. of (x—a). | 
it 
Ex. 9. The fluent of tal, ig —1a% — daz? —a?x + a3 X 
a——2x ‘ 
hyp. log. of (a—2). 
gtx 
Ex. 10. The fluent of 





is Lat —1axt-++la2e? —~a32-+ 








atx’ 
a4 Xhyp. log. (a-+2). 
3 mm n—i1 3 »n—22 
Ex. 11. The fluént of — OD Fae A Saal eae 
atgn—3 atnz’ n—1 n—2 





LT &e,  a® xh. 1. (aa), 


gt ame 


Ex. 12. 


342 HURON AND FLUENTS. 




















n=l n— 2 
8, chic 
ey &e. —a" xh. (ors) | 
n—1 Dy n—2 
“Be. 13; The fluent of ea is a a ie OT pee 
pein Nex] jam 2 % 
- 3 kc. +a" x he. =a). 
a2 
Ex. 14, The fluent of erty division) ic eran 


is, (by form 11.this vol. ) = CIF. arc B eit a and tang. a 


or g—1 paxcir. arc of rad. 1 and cosine — 





2 
eae = In like man- 
ner, | 

Ex. 15. The fluent ar os or of —x ties 


+ 7axh. I ate , by form 10. And 


,s—2 





a ise ax 


a? 





Ex. uss The fluent of ek =x eae 
hyp. log. ale , by the same form. 


a 
70. In like manner for the fluents of ae a” Thus, 
Ex. 17. The fluent fo = 227 —a2a7 + ae , Is 


by form, le? a? x +a? x cir. arc to rad. a and tang. 2, or. 
a2 —g2 g 
i783 —a?x+1a3 % cir. arc. to rad 1 and cosine’ —=———. And 


a? a2 

etx 2 x hee 

Ex. 18. The fluent of =— we cc aee is a aa? 
—Z x 


—, by form ke ~ Also 
Ex. 19. The fluent Ze ae es ft a? a re 





is —ig? —a%x + 1a°X bye: log. — 





mers. Is 


173 --a2x + 443 >< hyp. log. ~ era by form 10. 


71. And in general for the fluent of rere where nis any 


even positive number, by dividing till the powers of x in the 
numerator are exhausted, the fluents will be found as before. 
And first for the denominator x2-+-o2, as in 


Ex. 20. 


* 


FLUXIONS AND FLUENTS, 343 


Ex, 20. For the fluent of ra, = fby staal division ) 


a” 
» the 


a? -at ? 


number of terms in the quotient being in, and the remainder 


gn? jp ig 2gn—4 Jp + 4 28 ae oe + a ae pa LS aa 

a Vasa 

> eae? Viz. — or+ according as that number of terms is 
x 

odd or even. Hence, as before, the fluent 


4 gz! spatanr? 
Is —--+ &&.... 4 a?2 F a? &X arc to rad. 








n—1 n—3> : 
ni tans rp: ; 5 
~-+- &... ha? F 1a"! x 








a and tan 2, or 
11) 





ah ee ee 

2 ome V2 

arc to rad. 1 and cos. aes 

3 LN ; 
Ex. 21. In like manner, the fluent of = ees ee 
gl gagn 3g 4 gn5 ate 
ES ete) Weg ALAR gE OM Warne PRE 9,4) Ja™—!.x% h ] es, 
n= I n=—3 em eae ee awe an 
Bx, 22. And of - oeaitces te = 
she ° ss —_—— Agni. 
x nd o eons 1s Rel pa c. +. lq 


x hyp. log. ~ os : 

72. In a similar manner we are to proceed for the fluents 
of ao when x is any odd number, by dividing by the de- 
nominator inverted, till the first power of only be found 
in the remainder, ‘and when of course there will be one term 
less in the quotient than in the foregoing case, when n was 
an even number ; but in the present case the log. fluent of 
the remainder will be found by the 8th is in the table of 
fluents. 


Ex. 22, Thus, for the fluent-of — a ‘ at 2 is an a 
number, the quotient by division as 5 tletaee: is a"—2y — a24"—-4 
etate—5, — &et ee os anes of terms being 
m—1 





, and the remainder ei aa re . Therefore the fluent i is 
gr! a2x¢—3 Si 2 

mals Sy Ae e@eee oo ace _ Lgl h. 1, 2 2), 
es a ne 2 Neds eeia ae 








Ex. 93, ‘The fluent of a obtained in the same man- 
xe? — 


ner, and has the same terms, but the signs are all positive, 
and the remainder is -}- ta"—-! x hyp. log: ee — a2, 
. Ex. 24, 


344 FLUXIONS AND FLUENTS. 


Ex. 24. Also the fluent of a5 is still the same, but the 


signs are all negative, and the Ferhinded is — La”! X hyp. 
log. a2—2?. \ Hence also, 


_ Ex, 25. The fluent of ae 5 18 42? — 1a? X hyp. log. 
of x? +a2, 

Ex. 26, The fluent of sw is 3a? + 1a? ~*byp- log. 
of 2? —a?. 

rt» 

Ex. 27. The fluent of is 
log. of a2 —x3. 

Ex. 28. The fluent of —— 
hyp. log. 22--a2. 

Ex, 29. The fluent of 
log. 2? ~a?. 

Ex. 30. The fluent of te , is — Jas —lota?—1atX 


ig — 172 —1a2 X hyp. 


eg?’ 


as ae is 144 — latz? + 1a4 X 





, 18 }a4-+-10242-++-1a4 Xhyp. 


hyp. log. a? —2?. 
78. Ex. 31. In a similar manner may be found the 
geri, 


fluents of — ce where c is any whole Eeaitite wished by 


ARS ied sly 








dividing till the remainder be , which can always 


“be done, and the fluent of that reas will be had by the 
8th form in this vol. Thus, by dividing first by 2” -+ a”, the 
terms are, 2°"—*—1; — perpen typ 4 gingen—3n— 1, 
&c. till the last term be ae ras Or and the remainder. 
@2ng(c—d)n—1), heer nee 
gba atta” 
c, which is also the number of the terms in the quotient ; and 


therefore the fluent is 
gen—n aacn—2n gg 2@ngcn—Sn ale—2)%y7 ] 
x . pace bee —ale—I)n xX 
7? . 


when d is ==c—1, or 1 less than 











cn—n cn—2n | cn—3n n 


hyp. log. of a"-+-a". In pei nee 
Ex. 32, The fluent. of = 


as the former, but their ens sal +- or positive, and the re- 





= I has all the same terms. 


1 
mainder - : atc—1)" x hyp. log. of 2a". - Also in like manner, 
Exn33- 


FLUXIONS AND FLUENTS. 345 
Ex, 33. The fluent of ~——=-has all the very same terms, 


but all negative, and the remainder —* ale)" X hyp. log. 





of a™— 2x”. 
m—1), 1 gern ly 
Ex. 34. ‘The fluent of = a saa ped j is also the 
etx” 


same with the preceding, by substitut. for a”, and multiply: 


ing the whole series by the fraction : : 


74, When the numerator is compound, as well as the de- 
nominator, the expression may, in a similar manner by divi- 
sion, be reduced to like terms admitting of finite fluents. 
Thus for, 

i are, — bri z 

Ex. 35. To find the fluent of & wD ie Toke 

b d-+-b LC. : 
By division this becomes — bat Ot ta dp edit ; and its 


d “ta 

fluent — ry a + veut hyp. log. of © a+ x, 

a5. There are certain methods of fading fluents one from 
another, or of deducing the fluent of a proposed fluxion from 
another fluent previously known or found. There are hardly 
any general rules however that will suit all cases; but they 
mostly consist in assuming some quantity y in the form of a 
rectangle or product of two factors, which are such, that the 
one of them drawn into the fluxion of the other may be of the 
form of the proposed fluxion ; then taking the fluxion of the | 
assumed rectangle, there will thence be deduced a value of 
the proposed fluxion in terms that will often admit of finite 
fluents. ‘The manner in such cases will better appear from 
the following examples. 


Ex. 1. To find the fluent of —— aE 


Here it is obvious that if y be assumed = x ,/ (x? + a?), 
then one part of the fluxion of this product, viz. x X flux. of 
o/ (x? +-a?), will be of the same form as the fluxion proposed. 
Putting theref. the assumed rectangle eye / (x? + a?) into 


fluxions, itis y= x4/ (x?-++a?) -+- ——— ee paiy But as the 


Vou. Ws A5 former 


* Xap 








346 FLUXIONS AND FLUENTS. 


former part, viz. 24/ (x? ++ a2), does not agree with any of 
our preceding forms, which have been integrated, multiply it 
by 4/ (a? + a*), and subscribe the same as a denominator to 
the product, by which that part becomes 


g2ta2 , oe vPaebarte 
Var pay” . ai Paty} ; this united with the former part 
Qx2 a ax 
makes the whole y = ———__— ‘hones the 
wars» ba 
iven fluxion —— = }ly—1a? X —— , and its 
: ee ar Wet Fa?) 


fluent is therefore 1y —1a? Xf—— Y, range = 30 yf (2? + a? )— 


302 X hyp. log. of +a (a? +a), by the 12 form of fluents. 
‘Ex. 2. In like manner the fluent of WIE: Sassen be 
“ergy by the same 12th form, and is= . 


joy —a)+40, x ip Noe x+o/(a? — a7). 
Ex. 3. Also i in a similar manner, by the 13th form, the flu- 


2x 
ent of > oe “ay will be found from that of ——- Aaa _& ae ey’ and 
comes out — tx 4/ (a? — a?) -+- da X cir. arc to radius a ae 
sine x. | : | 





Ex. 4. In hke mannner, the fluent of —— A a ba?) will be 


2: 
- Here it is manifest that y 


found from that of —— a2)’ 


must be assumed =7%3,/(x? -+-a?), in order that one part of its 
fluxion. viz. « X flux. of ,/(«?-+a?) may agree with the pro- 
posed fluxion. Thus, by ree the fluxion, and reducing as 


before, the fluent Of F We ‘ om La) will be found = rats V (29+ 
Tei Fay 
Ex. 5. Thus also the fluent of 


v2» 


a?)—3a? X f-—— 





sa =i) is 70° 4/(x?--a7) 
+i Xf ae je ye 


Tena 


| Ex. 6. And cee 918 = 4234/(a? —x?) +302 x 


a? —«?) 
f 





22x 


In 


FLUXIONS AND FLUENTS. aa 


Yn like manner the student may find the fluents of 
xox See oh Anas : art 

a/ (x? ba?) f(x? Hat ay © J (02 a a? 

even number, each from the fluent of that which mediately 

precedes it in me Peres, by substituting for y as nee Thus 


the fluent of —— OG =F) “RSM (x3 +a) =~ Sa? Xf 


, where n is any 





"2 7 
JVarre) 

76. In lke manner we ‘may proceed for the series of simi- 
' lar expressions where the index of the power of x in the nu- 
merator is some odd number. 

Ex. 1. To find the fluent of errant Here aatebiini 
y=a? ,/(x?-+a?), and taking the fusion: one part of it will be 
similar to the fluxion proposed. Thus, y = 2r2 7 on gat: 

xo 
be hence at once the given fluxion 
1 Vere)! : Herta) 
=y —2xz ./(x?-++a?); theref. the yenaized fluent is y — f'. 
2x0 f(a? +a? )=22 AG +a?) — As ay ¥ the2d form 
of fluents. 


Ex. 2. In like manner the fluent. of ——— 


; Vv G pay 
22,/ (“2 —a?) ~2(a2 —a?)?, ate 3 
x8 
Ex. 3. And the fluent of ——— y Ge Teeawy~ Me — 2? au 
2 2 (a2 —2?)2, 
Ex. 4. To find the flu. ee ts bat) from that ob ea 
Here it is manifest we must assume y = x4 4/ (x3-+a?2). 
5a¥ a 4a2 x34 
Aer) Karte) 
and hence —— 7 ies ES a dthe fl 
nd henc Tetay : "Tera mnt e flu. 
isly~4a2 xf— HEEL e 1x4 ,/(a? +a?) +403 x f—__ Wert 
the fluent ah the latter part being as in ex. 1, ve 
In like manner the pronent may find the fluents of 
25 x 3s SAS Bo 
Jaa)” ae — a) 


lar way risk the fluents of —— 


ois) in fluxions and reduced gives y y= 


He may then proceedinasimi- . 


atx Sa re 
af (23 2 wa?) J (a2 aa): 


@ 


348° FLUXIONS AND FLUENTS.. 


“" x 
fers + 2). a? 
of the fluent of each preceding term in the series. 

77. In a similar manner may the process be for the fluents 
of ee series of aR 


, where n is any odd number, viz. always by means 


xu” 7 


WCET ey rice pier t J(a 2x)’ 
using the fluent of each Cr term in the series, as a 
part of pee next term, and knowing that the fluent of the 


‘first term 





ata ra given, by the 2d form of fluents, = 2 
(at x), of tive Same sign as x. 

it 1. To find the fluent of 63 C aie having given that of 
—~=24/(r-+a)=a suppose. Here it is ans we must 


+2V 





Seta) 
Kote SE Te 4 for then its flux. y= —*> Tere 
EY ie faz 
7 iGo Tat era | Vera eral 
hence —-— o : ay =2y—20a; and the required fluent is 2y— 
20a=22,/(4-+-a)—40,/(a+a)= (1#—2a) X2./(x-+a). 


Ip ae manner oe student will find the fluents of 


(t--a = —— +4 5 


Para paesy 
Ex. ie To find the fluent of 
J G =e +a)” 
Here y must be assumed = x? ,/ Canes : 





, having given that 


of —— 

Tera * 
for then taking the flu. and reducing, there is found ie aay 
=2y —4ap ; theref. f-— = = 24 — 408 =2124/(2--a)~4 


ap=2x VG ee a(x— 20) x Ff (ba) = = (9x? — 4ax-+48a? ) 

X P54/(a-+a). 

: In we same etnies the student will find the abi of 
te 


atote hice yA di eral, the flu. of ——_— 
Tarte any seamen iat Tray 
being given = he he will a the fluent of —_— ie a cay = oy a i 
OY (eam ao. 


78. In a similar way we might proceed to find the fluents of 
other classes of fluxions by means of other fluents in the ta- 
ble of forms :: as, for instance, such as az 4/(da—x?), x? 24/ 

dz 


FLUXIONS AND FLUENTS. 349 


(dz—2?), t3e4/(da—x?), &c. depending on the fluent of 
« 4/(dx—x?), the fluent of which, by the 16th tabular form, 
is the circular semisegment to diameter d and versed sine z, 
or the half or trilineal segment contained by an arc with its 
right sine, and versed sine, the diameter being d. | 
Beet, Putting then said semiseg. or fluent of x J (dx —2?) 
=a, to find the fluent of xi ./(de—a?). Here assuming y= 


(da ~22)?, and taking the fluxions, they are y =2 (dir— 2x2 ) 
NAS cea hence 2a 4/ (da —a? )=1di ./ (dx ~ 2?) — ay = 
Ada —1y; therefore the required fluent faa J (dx — x2), is 


Ada—ly=1da—} (dx—x?)? =B suppose. 

Ex. 2. To find the fluent of 273 J (dx—a?), having that 
of xa 4/(de—a?) given =8. Here assuming y=x (dx—z2*), 
then taking the fluxions, and reducing, there results y= 
(idxz — 432; Jef (de =a") 5 hence 2? x 4/ (dx ~ x?)=idra sf 
(dy—ax?) ~11 =dzp— iy, ie flu. theref. of 22 4 / (dam x? ) 
is 5dg aoa —inr(dx— 22)?, 

Ex. 3. In the same manner the series may be continued to 
any she ; so that in general, the flu. of x*~1,/ (dx —2?) 
being given =c, then the next, or the flu. of z"2 ,/(dzr—«?) 


widen aye 





1 
1 
a+2 #7 n+2 
79. To find the fluent of such expressions as —— 


a/ a/ (x? ara 
a case not included in the table of forms. 
Put the proposed radical ,/ (x? = 2ax) == <z, or x? + 2az 
== 27; then, completing the square, 1? + 2ax 4a? = 2?--a?, 
and the root ist ta=/ ae --a3)),. The ae of this is 


c= ae Satine eref. —— WCE E8az)~ Tartar)’ the flu: 
ent of which, by the 12th form, is the oe log. of z + 4/ 
(22--a*)=hyp. log. of +a -+ ./ (x? +2az), the fluent re- 
quired. 


Ex, 2. To find now the fluent of having 


Lx 
Abie 22 
given, by the above example, the fluent of —— WCE + BC a 
suppose. Assume Sf {88+ ae) = y; ve its fuxion is 

Lx ay 
Je +22) Tai ttaa) 9) yeetaay 
—~aa; the fluent of which i isy — GA =f (a? + 2ar) — aa, 
the fluent rah 


=y; theref. —— 


350 FLUXIONS AND FLUENTS. 


Ex, 3. Thus also, this fluent of —— We FE oie 75) being: given ‘ 


the flu. of the next in the series, or—— will be found, 
V (a Dan 12) 

by assuming 24/ (at 2ax) = = 4; and so on for any other 
n—1 > é 

of the same form. — As, if the fluent of ——~ be given 
(ai 42a) | 

; then, by assuming a1 (x? + 2ax) = y, the fluent 

x" x 2Qn—: 


| 1 
9/22 +208) sald ACh Ruut os 





ac. 
n : 


Ex. 4. In like manner, the fluent of —— 
ae =2e5) 


JV (a? 
found ; and thus the series may be conta exactly as in 
the 3d ex. only taking — Ras for ++. 2ar. 


being 


given, as in the first eciauipte) that of —— “Dax may be 


80. Again, having given the fluent of » which 


x 
a/ (20a —2?') 
is — X circular arc to radius a and versed sine x, the fluents 
Le x? x lg 5a | 
of (2ax—2?)’ ./(Qax—ax2)’? "/ (2an— a?) 
ciate by the same method of SeSEN Es Thus, 





, may be 


Ex. 1.° For the fluent of — 3 ray etme (2ax—x? ) 


==y; the required fluent will be found = Si ae 
er arc to radius @ and vers. 2. 





Ex, 2: In like manner the fluent of a 
/ (20% = aaa) 
2 axe 3a-+-x 
—x?)=3 ae ps st 
eye rree “ay7 La,/(2an—x ve aa "VY (20x a), 


where a denotes the arc mentioned i in the Test crane. 
a 


Ez... 3. >And. in general the fluent of ———. Wars ee ak). 


2n—} 1 
bacanieTiteir y eeagage ee —~x? here is the fluent f 
rae = a Wf (2ax—x*), where c 1 nt o 
gl, 
th term in the series. 
Haan aay e next preceding ter i 


8t. pia also, the fluent of + J (x—a) being given, = 


(2— a)?, by the 2d form, the fluents of rz ./(« — a), re 

(a—a); &ic.. ~ 2" 4/ (a—a), may be found. And in gene- 

ral, if the fluent of x"—'; 4/ (x — a) =c be given; then by 
- assuming 


FLUXIONS AND FLUENTS. 3651 


assuming s(n — -a)" fish the fluent of a2 ./ («—«a) is found= 





an+3- ie a)! +550 k 
YER Sareea | 
82. Also, given the fluent of (z —a)"z which is uy 
(x—a)”"*! by the 2d form, the fluents of the series @aa)nxi, . 
(a—~a)"n?7 &e.. .. (w—a)2"j can be found. And in ge- 


neral, the fluent of (x —a@)”x"—1z being given =c ; then by 
assuming (« —a)”*!z" = y, the fluent of (a—a)nars is found 
Ce te a)"*1--nae 

. m+n+1 
Also, by the same way of continuation, the fluents of 
ae f (asx) and of xz (deb”)” may be found. 

83. When the fluxional expression contains a trinomial 
quantity, as ,/ (b + cx +- «?), this may be reduced to a bino- 
“mnial, by substituting another letter for the unknown one z, 
‘connected with half the coefficient of the middle term with its 
sign. Thus, putz =«+1e: then z?=x2?-+-cx-+1c?; theref. 

22 — 1¢2 = x2 hex, and 24 -- b—1c? =“? Minted the giver 
trinomial which is = z? -+ a?, by patting a? == b —1¢2, 


Ex. 1. To find the fluent of ———— ey aera 

Herez=2+ 2; then 2? =? + 42 + 4, and 22? +1=— 
5 + 42 + «?, also» =z; theref. the. ‘proposed fluxion re-. 
duces to ire A ; the fluent of which, by the 12th form in 
this vol. is 3 hyp. log. of Bite Af (1 + z) = hyp. log, a2 
+ (5+ 40+ 2°). 

oe 2. To tind the fluent of z 4/(b--ex-+-dx? =x VE Yeah 


G+ Sate), 

Hers assuming ato = =z; thenzx—z, and the proposed 
flux. reduces to z axe + 2 = JS 1X ./ (22-402), 
putting a? for — 7 -ie 7a? and the fluent will be found by a simi,, 


r 





lar process to that employed in ex. 1 art. 75. 
Ex: 3, In dike manner, for the flu. of «"—12 Jb “th ae 


dx2”), assuming ee =z, nx” —\y=7, and gm—le — hence 
a oe : an Anas ma /d Kf (ah 
cabal dX (zt - a a=) Jf AX f(z? £a?), putting = a 


qQ2 = 


352 MAXIMA AND MINIMA. 


6b ¢ 1 
a? == ~~; hence the given Piston becomes — ae dX Vv 
(2? a? ), and its fluent as in the last example. 


: a" ly 
Ex». 4.. Also, for the fluent of ——— ae raery Fares ) 
LD 


1 
==z, then the fluxion may be meee to the form — Tn aber veer 


; assume a7-+ — 
2? tas 


and the fluent found as before. 

So far on this subject may suffice on the present occasion. 
But the student who may wish to see more on. this branch, 
may profitably consult Mr. Dealtry’s very methodical and © 
ingenious treatise on Fluxions, lately published, from which 
several of the foregoing cases and examples have been taken 
or imitated. 


—— 


OF MAXIMA AND MINIMA ; OR, THE GREATEST 
AND LEAST MAGNITUDE OF VARIABLE OR FLOW. 


ING QUANTITIES. 


84, Maximum, denotes the greatest state or quantity atiain: 
able in any given case, or the greatest value of a variable 
quantity : by which it stands opposed to Minimum, which is 
the least possible quantity in any case. 

Thus the expression or sum a? --da, evidently.increases as 
x, or the term 6x, increases; therefore the given expression 
will be the greatest, or a maximum, when ~ is the greatest, or 
infinite: and the same expression will be a minimum, or the 
least, when x is the least. or nothing. 

Again i in the algebraic expression a? — ba, where a and b 
denote constant or invariable quantities, and x a flowing or 
variable one. Now, it is evident that the value of this re- 
mainder or difference, a2? —bz, will increase, as the term bz, 
or as x, decreases ; therefore the former will be the greatest, 
when the latter is the smallest ; that is a? —6z is a maximum, 
when «x is the least, or nothing at all; and the difference is 
the least, when x is the greatest. 

85. Some variable quantities increase continually ; ; and so 
have no maximum, but what is infinite. Others again decrease 
continually ; and so have no minimum, but what is of no mag- 
nitude, or nothing. But, on the other hand, some variable. 
quantities increase only to a certain finite magnitude, called 
their Maximum, or greatest state and after that they decrease 


again. While others decrease to a certain finite magnitude, 
called 


MAXIMA AND MINIMA, 353 
called their Minimum, or least state, and afterwards increase 
again, And lastly, some quantities have several maxima and 


minima. 


CDE BO 












AB GH dC SACS aaa Blba a 6. Saas 


Thus, for example, the ordinate Bc of the sparecl or ° 
such like curve, flowing along the axis an from the vertex a, 
continually increases, and has no limit or maximum. ~ And the 
ordinate cr of the curve ern, flowing from £ towards _n, con- 
tinually decreases to nothing when it arrives at the point u. 
But in the circle 11m, the ordinate only i increases to a certain 
magnitude, namely, the radius, when it arrives at the middle 
as at kt, which is its maximum; and after that it decreases 
again to nothing, at the point m. And in the curve nog, the 
ordinate, decreases only to the position op, where it is least, 
or a Minimum ; and after that it continually increases towards @. 
But in the curve rsu, &c. the ordinates have several maxima, 
as ST, wx, and several minima, as vu, yz, &c. 

86. Now, because the fluxion of a variable quantity, is the 
rate of its increase or decrease: and because the maximum 
or minimum of a quantity neither increases nor decreases, at 
those points or states,; therefore such maximum or minimum 
has’no fluxion, or the fluxion is then equal & to m OEMIOE: From 
which we have the following rule. 


To find the Maximum or Minimum. | 


87. From the nature of the question or problem, find an 
algebraical expression for the value, or general state of the 
quantity whose maximum or minimum is required ; then take 
the fluxion of that expression, and put it equal to nothing ; 
from which equation, by dividing by, or leaving out, the flux- 
ional letter and other common quantities, and performing 
other proper reductions, as in common algebra, the value of 
the unknown quantity will be obtained, wife acre git the point 
of the maximum or minimum. | 

So, if it be required to find the maximum ‘state of the 
compound expression 100zx — 5x° +c, or the value of x 
when 100% — 5a? + c isa maximum. The fluxion of this: 
expression is 100z — 10x; == 0; which being made = 0, 
and divided by 102, the equation is 10 —- s = 0; and hence 
z= 10. Thatis, the value of x is 10, when the expression 

Vor. I. A6 1002 


354 mene, “BLUKIONS, 


1002 -— 52? + cis the greatest. As is easily tried: for if 16 
be substituted for x, in that expression, it becomesstc+500 : 
but if, for « there be substituted any other number, whether 
greater or lesser than 10, that expression wiil always be found 
to be less than + c + 500, which is therefore its greatest 
possible value, as its maximum. 

88. It is evident, that if a maximum or minimum be any 
way compounded with, or operated on, by a given constant 
quantity, the result will still be a maximum or minimum. 
That is, if a maximum or minimum be increased, or de- 
creased, or multiplied, or divided, by a given quantity, or 
any given power or root of it be taken; the result will still 
be a maximum or minimum. Thus, ifa bea weal: or mi- 


nimum, then also is x -+ a, or x—aQ, Or ax, or —, OY 2“,, OF 


/a, still a maximum or minimum. Also, the Gaerne of 
the same will be a maximum ora minimum. And therefore, 
if any proposed maximum or minimum can be made simpler 
by performing any of these operations, it 2s better to do so, 
before the expression is put into fluxions. 

89. When the expression for'a maximum or minimum con- 
tains several variable letters or quantities; take the fluxion 
of it as often as there are variable letters ; supposing first one 
of them only to flow, and the rest to be constant; then an- 
other only to flow, and the rest constant ; and so on for all of 
them: then putting each of these fluxions = 0, there will be 
as many equations as unknown letters, from which these may 
be all determined. For the fluxion of the expression must be 
equal to nothing in each of these cases ; otherwise the ex- 
pression might become greater. or less, without altering the 
values uf the other letters, which are considered as constant. 

So, if it be required to find the values of w and y when 
Ax* —xy+2y is aminimum. | ‘Then we have, 

First, - 8x0 — ay =0, and 8e—y = 0, or y= 8x. 

Secondly, 2y — xy = 0, and 2—a=0, or « = 2. 

And hence yor 8x = 16*. 





* The numbers here found, viz. 2 aa 16 for # and y, do not render the propos: 
ed formula 4x? —xy4-2y either a maximum or minimum. 
| lf x = 1, y=10, then 42? —ayt2y = 14, 
xx: 2, y= 16, then 427 —xy-+ 2y — 16, 
x= 3, y=10, then 427 —ryf-2y = 26. 
Th cae put x= 2+e, y= 16 +f, and by substitution we have 
4a? —wy =f 2y = 16-1.4e? —ef. é 
It is evident that if 40? _ef were always positive, whatever values, positive or 


MAXIMA AND. MINIMA.. 355 
90. To find whether a proposed quantity admits of a Maximum 
or a Minimum. 


Every algebraic expression does not admit of a maximum 
or minimum, properly so called ; hs it may either increase 











negative we assign to ¢ and f, the least value of 16 + 4e?—ef would be found 
by taking e and f each==0; or if 4e* — ef were always a negative quantity, 
whatever positive or negative values we give to e and_//, then 16 4c? —ef would 
be a maximum by taking e and f each=0: but if by changing the signs of e and 
A, or their ratio, the formula 4e” —efcan be changed from positive to negative or 
from negative to positive, it is clear that the quantity 16 +4e” —ef cannot be 
either a maximum or minimum by making e and f each = 0. 


Now 4e? — e ae? (4 = and 4—*= can evidently be made positive or nega- 
tive at pleasure. For example, if e= i" f=1, we have 4e7 —ef =3. In this 
case »==2+4-1=3, y= 16-+ 1=17, and therefore An? — ay $2y=164+3=19. 
Ife = 1, f=5, then tt ett Here x= 2 41 =3, y= 1645 =21, and 


4x0? — wy Q2y = 16 —1= 15. . 
Thus we see the reason why the proposed formula cannot be a maximum or 
minimum when «== 2, y= 16. 

Let any expression in. and y be proposed to be a maximum or minimum. 
Find by the rule in the text the values a and }, of « and y, corresponding to the 
Maximum or minimum. Put «=a-+e, y= b+, and by substituting a+ e, and 
b+-f for « and y, in the expression propcera to be made a maximum .or minimum, 


let ce 2 4 pef+-af? 


be that part of the proposed expression which involves the second powers of ¢ and 


FJ, c, D, and cacy obtained. in terms ofa and b, | Put=—=u and we have 


St 
ce? +P vef-+FE ‘ a fe FP 4 24,7 4-4cpu-ev? “-4cz—p? t. 


2 


Put 4cu -- p =a, and if 4 ce—p* =p” be positive, put it 7 and we have 


2 2 2 
ce* 4. nef + nf ee SoF 45 : : 
c 
It is now evident that when 4ck—p? js positive, the quantity ce? -nef-puf? 
cannot be made to change its sign, but est remain of the same sign with the 


coefficient c, and consequently when 4cz— D7 is a positive quantity, the proposed 
expression is a maximum or minimum asco as C Is negative or affirmative. 
When 4cz— pv? is negative, put it —=— my , and we have 


ce 2 + nef yp nf? = Sa —B2}. 
Now since a =2cu--p, in which u is neue and may be taken at pleasure, it 
is evident that a may become of any value whatever, while 8? remains constant, 
and therefore by taking different values of u the quantity a? — g? may be both 
positive and negative, and of course ce? +pnef+nf A may change its sign by as- 
signing different values to; and therefore when 4cz— p? is negative, the pro- 


posed formula can neither be a maximum nor minimum: 
In the 


856  » FLUXIONS. 


continually to ‘infinity, or decrease continually to nothing ; and 
in both these cases there is neither a proper maximum nor 
minimum ; for the true maximum is that finite value to which 
an expression increases, and after which it decreases again : 
and the minimum is that finite value to which the expression 
decreases, and after that it increases again.. Therefore, when 
the expression admits of a maximum, its. fluxion is positive 
before the point, and negative after it ; but when it admits of — 
a minimum, its fluxion is negative before, and positive after it. 
Hence then, taking the fluxion of the expression a little be- 
fore the fluxion is equal to nothing, and again a little after the 
same ; if the former fluxion be positive, and the latter nega- 
tive, the middle state 1s a maximum, but if the former fluxion 
be negative, and the latter positive, the middle state is mini- 
mum. 

So, if we would find the quantity ¢ ax — x? a maximum or 
minimum ; make its fluxion equal to nothing, that is, - - - 
ae — 2xx = 0, or (a —2xr)z = 0; dividing by z, gives 
a— 2x = 0,or x = ia at that state. Now, if in the fluxion 
(a —.2r) x the value of « be taken rather less that its true 
valae 1a, that fluxion will evidently be positive ; but if 2 be 
taken emda lant greater than 1a the value of a— 2x, and 
consequently of the fluxion is as ‘evidently negative. There- 
fore, the fluxion of az — x? being positive before, and nega- 
tive after the state when its fluxion is = 0, it follows that at 
this state the expression is not a minimum but a maximum. 

Again, taking the expression «? — az?, its fluxion 327; — 
2are, =(3x—2a)xx= 0; this divided by rx gives 3r—2a=0, 








In the example in the text we have ce~ eee neff? ide? ef: 502% 
Here c—=4, p==—1, u =e, hence 4zx — np? ==—, and therefore the formula 
4x* Quy 4.2y has neither maximum nor minimum. 

If in the substitution of a--e and 5+4-f for x, and y.in the given expression, 
the coefficients c, D, # of the second order vanish, the increment will depend on the 
terms of the third order, in which case the quantity can have neither maximum nor 
minimum. Should the terms of the third order also vanish, we must have recourse 
to the terms of the fourth order, the’ sum of which as in the case of the second 
powers must be incapable of changing its sign that there may be a maximum. 
In general, that the proposed formula may be a maximum or minimum, the incre- 
ment must have its lowest powers of ¢ and f of an even order, and the sum of 
all these even terms must be incapable of changing its sign; or, which amounts 
to the same thing, the sum of these terms equated to 0, must give only i imaginary 
value for each of the quantities e, f. 

‘The same rule is equally eee when there are any numbers of independ- 
ent variables, x, y, 2, &c. ; 

The coefficient c, p, x, &c. may be obtained with the greatest ease, by ns 
successive fluxions according to. the method of partial differentials. 

apd 


MAXIMA AND MINIMA. 357 


and ¢=2a, its true value when the fluxion of a? — az? is equal 
to nothing. But now to know whether the given expression 
be a maximum ora minimum at that time, take x a little less 
than 2a in the value of the fluxion (32 — Qa)a% , and this will 
evidently be negative ; and again, taking « a little more than 
2a, the value of 3x—2a or of the luxion, is as evidently po- 
sitive. Therefore the fluxion of x =~ ax? being negative be- 
fore that fluxion is =0, and pools after it, it follows that in 
this state the quantity x? —ax? admits of a minimum, but not of » 
a maximum. . 


91. SOME EXAMPLES FOR. PRACTICE. 


Exam. 1. To divide a line, or any other given quantity a, 
into two parts, so that their rectangle or product may be the 
greatest possible. 

Exam. 2. To divide the given quantity a into two parts 
such that the product of the m power of one, by the n power 
of the other, may be a maximum. 

Exam. 3. . To divide the given quantity a into three parts 
such that the continual product of them all may be a maxi- 
mum. 

Exam. 4. To divide the given quantity a into three parts 
such, that the continual product of the Ist, the square of the 
2d, and the cube of the 3d, may be a maximum. 

Exam. 5. To determine a fraction such that the difference 
between its m power and ” power shall be the greatest pos- 
sible. : 

Exam. 6. To divide the number 80 into two such parts 
and y, that 2a2-+-xy+3y? may be a minimum. 

Exam. 7. To find the greatest rectangle that can be in- 
scribed in a given right-angled triangle. 

Exam. 8. To find the greatest rectangle that can be in- 
scribed in the quadrant of a given circle. 

Exam. 9. -To find the least right-angled triangle that can 
circumscribe the quadrant of a given circle. . 

Exam. 10. . To find the greatest rectangle inscribed in, , and 
the least isosceles triangle circumscribed about, a given sem!- 
ellipse. 

Exam. 11. To determine the same for a given parabola. 

Exam. 12. To determine the same fora given hyperbola. 

Exam. 13. To inscribe the greatest cylinder in a given 
cone; orto cut the greatest cylinder out of a given cone. 

Exam. 14. To determine the dimensions of a rectangular 
cistern, capable of containing a given quantity a of water, so 
as to be lined with lead at the least possible expense. 

Exam. 15. Required the dimensions ofa eyennce! ie 

ards 


358 | TANGENTS. 


kard, to hold. one quart of ate measure, that can be made of 
the least possible quantity of silver, of a given thickness. 
_ Exam. 16. To cut the greatest parabola from a given cone. | 
Exam. 17. To cut the greatest ellipse from a given cone. 
Exam. 18. To find the value of z when 2* is a minimum. 


———— 


THE METHOD OF TANGENTS; OR, TO DRAW 
TANGENTS TO CURVES. | 


92. ‘The Method of Tangents, is a method of determin- 
ing the quantity of the tangent and subtangent of any algebra- 
ic curve ; the equation of the curve being given. | Or, vice 
versa, the nature of the curve, from the tangent given. 

If sz be any curve, and E be | 
any point in it, to which itis re- 
quired to draw a tangent TE. 
Draw the ordinate erp: then if 
we can determine the subtan- 
gent rp, limited between the or- 
dinate and tangent, in the axis 
produced, by joining the points, 
r, E, the lne re will be the tan- 
gent sought. 

93. Let dae be another ordinate, indefinitely near to DE, 
meeting the curve, or tangent produced in e; and let re he 
parallel to the axis ap. Then is the elementary Hien: Eca 
similar to the triangle rpg ; and 





therefore - e@:@E::ED: DT. 

But - = e@:ae:: flux. ep ; flux. ap. 

Therefore - flux. ep : flux. aD :: DE: DT. 
Yor 


Thatis,- - y: pay ya OE ee or! 


which is therefore the general value of the subtangent sought ; 
where ~ is the absciss ap, and y the ordinate. pn. 
Hence we have this general rule. 


- GENERAL RULE, 

94. By ‘means of the given equation of the curve, when 

put into fluxions, find the value of either z or y or of ge 
y. 

which value substitute for it in the expression pr =o 


y 
and, 


RECTIFICATIONS. 359 


and, when reduced to its simplest terms, it will be the value 
of the eubianeent sougnt. | 


EXAMPLES. 


ene 1. Let the proposed curve be that which is defined, 
or expressed by the equation ax? +-2y? —y?=0. 
Here the fluxion of the equation of the curve is 
2are tye +2ryy —3y?y=0; then, by transposition, 
Zara +y2 2 =3y?y — 2ryy ; and hence, by division, 
Oy? = Say. Ye Reh ty? 
7 eae ge consequently = ei paecua tt 
which is the value of the subtangent Tp sought. 
- Exam. 2. To draw a tangeut to a circle ; the equation of 
which is ox—ax?=y? ; where zx is the absciss, y the ordinate, 
and a the diameter. | 
Exam. 3. To draw a tangent to a parabola ; its equation 
being ax = y? ; where a denotes the parameter of the axis. 
Exam. 4. To draw a tangent to an ellipse ; its equation 
being c? tae? aes ; where-a and ¢ are the two axes. 
Exam. 5, To draw a tangent to an hyperbola; its equation 
being c? (aa-toa \=a7y2 ; where a and ¢ are the two axes. 
Exam. 6. To draw a tangent to the hyperbola referred to 
the asymptote as an axis ; its equation being ry=a? ; where 
a? denotes the rectangle of the absciss and ordinate answer- 
ing to the vertex of the curve. 





=a 


OF RECTIFICATIONS ; OR, TO FIND THE 
LENGTHS OF CURVE LINES. 


95. Recrirication, is the finding the length of a curve 
line, or finding a right line equal to a proposed curve. 

By art 10 it appears, that the 
elementary triangle nae, formed by 
the increments of the absciss, ordinate, 
and curve, is a right-angled triangle, 
of which the increment of the curve is . 
the hypothenuse : and therefore the PR ar ARDC 
square of the latter is equal to the sum 
of the squares of the two former ; that is, re?=£a?--ae*. 
Or, substituting, for the increments. their proportional flux- 
ions, it is 2z=2e + yy, Or 2 =f a? + y? 72; where z denotes 


any curve line ar, x its absciss ap, and y its ordinate px. 
Hence this rule. 








RULE. 


360 | FLUXIONS. 
| RULE. 
96. From the given en trilon of the curve put into fiaeiane. 


find the value of x? or y*, which value substitute instead of 


it in the equation z = /x?+y? ; then the fluents, being tak- 
en, will give the value of z, or the length of the curve, in 


Gee of the absciss or ordinate. 





EXAMPLES. | 


Exam. 1. To find the length of the arc of a circle, interms 
of the sine, versed sine, tangent, and secant. 

The equation of the circle may be expressed in terms of 
the radius, and either the sine, or the versed sine. or tangent, 
or secant, &c. of an arc. Let therefore the radius of the 
circle be ca or ce=r, the versed sine ap (of the arc az) =z 
the right sine pp = y, the tangent re=1, and the secant cr=s, 
then, by the nature of the circle, there arise these equations, 
viz. fos merge 

nit? $?—7? 
y= =r — x? = ae — = Stat : 

Then, by means of the fluxions of these enidu tions: with 
the general fluxional equation z?== %?-+ y?, are obtained the 
following fluxional forms, for the fluxivn of the curve; the 
fluent of any one of which will be the curve itself; viz. 

Te Mes OE oe La 725 


a epee te 1H OT Oana 

f2re—ae ofr? —y? 7 1/32 =r? 

Hence the value of the pee ‘from the fluent of each of 

these, gives the four following forms, in series, viz. putting 
d==2r the diameter, thecurveis — 


3x2 3.523 
rine hh 2. ear 252 A, ade. 4.6.7d3 


- 3.64% 
oe a & 
ae Sr? +33 4. eae Tere ody 
AM 6 t8 


sa T Egat inet 9p ' 
s—r, 8? — 73 3(s5 78 
= toe bape 

Now, itis evident that the simplest of these series, is the 
third in order, or that which is expressed in terms of the tan- 
gent, . That form will therefore be the fittest to calculate an 
example by in numbers, . And for this purpose it will be con- 
venient to assume some arc whose tangent. or at least the 
square of it, is known to be some small simple number. Now, 
the arc of 45 degrees, it is known, has its tangent equal to 
‘+ «the 














gee ye, 


=(1— so &c.) t, 








FLUXIONS. 361 


the radius ; and therefore, taking the radius r=1, and conse- 
quently the tangent of 45°, or t==1 also, in this case the arc 
of 45° to the radius 1, or the arc of the quadrant to the dia- 
meter 1, will be equal to the infinite series 1 — 2 + 4 ~— 3+- 
1 &c. - | 

” But as this series converges very slowly, it will be proper 
to take some smaller arc, that the series may converge faster ; 
such as the arc, of 30 degrees, the tangent of which is =,/2, 
or its square t?2==1 : which being substituted in the series, 
the length of the arc of 30° comes out. - - - +--+ - = 

1 1 1 1 

(I ye ae MACY — &.) v5: Hence, to .com- 
pute these terms in decimal numbers, after the first, the suc- 
ceeding terms will be found by dividing, always by 3, and these 
quotients again by the absolute number 3, 5, 7, 9, &c. ; and 
lastly, adding every other term together, into two sums, the 
one the sum of the positive terms, and the other the sum of 
the negative ones: then lastly, the one sum taken from the 
other leaves the length of the arc of 30degrees ; which being 
the 12th part of the whole circumference when the radius is 
i, or the 6th part when the diameter is 1, consequently 6 times 
that arc will be the length of the whole circumference to the 
diameter 1. Therefore multiplying the first term ,/i by 6, 
the product is ,/ 12 = 3°4641016 ; and hence the operation 
will be conveniently made as follows : - 








+Terms. | — Terms. 
1 ) 3-4641016 3°4641016 —. 
3 ) 1:1547005 0°3849002 
5) 3849002 ( 769800 . 
7) 1283001 183286 
9.) 427667 47519 
1}°) |. 142556:..¢ 12960 
13 47519 ( 3655 | 
15 , 15840 ( | 1056 
17 5280 ( 311 ; 
19 1760, ( oS 
21) 587 ( 28 
23 196 8 
25 65 3 
27") + Ste ¢ ' 1 
+-3- 5462332 —0°4046406 
— 0:4046406 








So that at last 3°1415926 is the whole circum. 
¥ : tothe diameter 1. 
Vox. IT, A’ ' Exam, 2, 


eT 





362 QUADRATURES. 


Exam. 2. To find the length of a parabola. 

Exam. 3. To find the length of the semicubical parabola, 
whose equation is ax2=y?. 

_ Exam. 4. To find the length of a an elliptical curve. 

Exam..5. To find the nec of an byperbolic curve. 


ps 


OF QUADRATURBS OR, FINDING THE AREAS 
OF CURVES. 


97. The Quadrature of Curves, is the measuring their 
areas, or finding a square, or other right-lined space, equal 
toa proposed curvilineal one. 

By art. 9 it appears, that any flowing 
quantity being drawn into the fluxion of 
the line along which it flows, or in the 
direction of its motion, there is produc- 
ed the fluxion of the quantity generat- 
ed by the flowing. That is, pd X pe 
or yx is the fluxion of the © are@ ADE. 
Hence en rule. 





RULE. 


98. From the given equation of the curve, find the value 
either of z or of y ; which value substitute instead of it in the 
expression yr ; then the fluent of that expression, being taken, 
will be the area of the curve sought.. 


EXAMPLES. 


Exam. 1..To find the area of the comtaGn pt aye 

The equation of the parabola being ax = y? ; where a is 
the parameter, « the absciss ap, or part of the ri and y the 
ordinate bE. 

From the equation of the curve is foundy =,/axr. This 
substituted in the general fluxion of the area yx gives z4/ ax 


or ata?z oe Paxton of the parabolic area; and the fluent of 


this, or 2q?4? = = 29 ,/ ax = 2ay, is the area of the parabola 
ADE, and which. is therefore equal to 3 of its circumscribing 
rectangle. 

Exam: 2. To square the rele: or find its area. 

The equation ‘of the circle being y? = az —z?, ory = 


Jae Bs | Phere se a is the diameter; by culntuben: the 
. one 


SURFACES. _ 363 


general fluxion of the area yx , becomes ~ ,/ax — x*, for the 
Auxion of the circular.area, But as the fluent of this cannot 
be found in finite terms, the quantity ./ax — x? is thrown 


into a series, by extracting the root, and then the fluxion of 
the area becomes a 
7 B2 1.323 1.3.524 


‘ x 3 
fad OA oh dea 7) Bae Sk Oat ee 
and then the fluent of every term being taken, it gives 
AU Ee con OO Saas ane 
te NAGS ae Taga abled? 4.6.8.1 1a" 
for the general expression of the semisegment ave. 
And when the point p arrives at the extremity of the dia- 
meter, then the space becomes a semicircle, and zx =a; and 
then the series above becomes barely 
32 1 I 1.3 1.3.5. 
a8 (Ge ee a bt) 
3.5 4.7 4.6.9. 4,6.8.11 
for the area of the semicircle whose diameter is a. 
Exam. 3. To find the area of any parabola, whose equation - 
is a@2%= yn, ' 
Exam. 4, To find the area of an ellipse. 
Exam. 5. To find the area of an hyperbola. 
Exam. 6, To find the area between the curve and asymp- 
tote of an hyperbola. , 


Exam. 7. To find the like area in any other hyperbola 
whose general equation is 2”y"==a™', <i 





ge 


TO FIND THE SURFACES OF SOLIDS. 


99. In the solid formed by the rota- 
tion of any curve about its axis, the sur- 
face may be considered as generated by 
the circumference of an expanding cir- 
cle, moving perpendicularly along the 
axis, but the expanding circumference 
moving along the arc or curve of the _ , 
solid. Therefore, as the fluxion of any generated quantity is 
produced by drawing the generating quantity into the fluxion 
of the line or direction in which it moves, the fluxion of the 
surface will be found by drawing the circumference of the 
generating circle into the fluxion of the curve. That is, the 
fluxion of the surface, gan, is equal to Ag drawn into the cir- 
cumference scer, whose radius is the ordinate DE. 

100. But if be = 3.1416, the circumference of a circle 

whose 





364 FLUXIONS. 


whose diameter is 1, x = ap the absciss, y = vx the ordi- 
nate, and z = az the curve; then 2y = the diameter be, 
and 2cy = the circumference scer; also, an = 2 = 
/xty?: therefore 2cyz or 2cy ./z2+72 is the fluxion of 
the surface. And consequently if, from the given equation 
of the curve, the value of x or y be found, and substituted in 
this expression 2cy ./x?+y?, the fluent of the expression 
being then taken, will be the surface of the solid required. 
; EXAMPLES. 

Exam. 1. To find the surface of a sphere, or of any seg- 
ment. , 

In this case, az is a circular arc, whose equation is y2 =ax 
—2x?, or y=\faxr—x?. . | | 





O—2%  . G=—2zx, 
2af ax — x2 _ ay 
a2?—4ar+4x2. 0 a? —4y?. a chineaucntly ack 

Aye ied 7 Pr qt yx 
Je a4 gc? Sree) ley 
fa aos 2442 _, 
Digg e ns eM ay 

This value of z, the fluxion of a circular arc, may be found 
more easily thus: In the fig. to art. 95, the two triangles epc, 
Eae are equiangular, being each of them equiangular to the 
triangle erc ; conseq. EP: EC :: Ea: ee, thatis, - - - - 


The fluxion of this gives y = 


hence y ?= 








pads Seek ee 
y:ia::2:z=—<, the same as before. 


The value z of being found, by substitution is obtained 2cyz 
s=acx for the fluxion of the spherical surface generated by 
the circular arc in revolving about the diameter ap. Andthe 
fluent of this gives aca for the said surface of the spherical 
segment BAE. ; 

But ac is equal to the whole circumference of the gene- 
rating circle; and therefore it follows, that the surface of any 
spherical segment, is equal to the same circumference ef the 
generating circle, drawn into a or ap, the height of the seg- 
ment. 

Also when « or ap becomes equal to the whole diameter a, 
the expression acz becomes aca or ca*, or 4 times the area of 
the generating circle, for the surface of the whole sphere. 

_ And these agree with the rules before found in Mensuration 
of Solids. Pr dae ; 

Exam. 2. To find the surface of a spheroid. 

Exam. 3. To find the surface of a paraboloid. 

Exam. 4. ‘To find the surface of an hyperboloid. | 

; TO 


. 


LOGARITHMS. 365 


TO FIND. THE CONTENTS OF SOLIDS, 


101. Any solid which is formed by the revolution of a curve 
about its axis (see last fig.). may also be conceived to be ge- 
nerated by the motion of the plane of an expanding circle, 
moving perpendicularly along the axis. And therefore the 
area of that circle being drawn into the fluxion of the axis, 
will produce the fluxion of the solid. That is, ap X area of 
the circle scr, whose radius is DE, or diameter Bg, Is the flux- 
ion of the solid, by art. 9. 

102. Hence, if ap==x, pe= y, ¢ = 3'1416; because cy? is 
equal to the area of the circle scr: therefore cy?z is the 
fluxion of the solid. Consequently if, from the given equa- 
tion of the curve, the value of either y? or z be found, and 
that value substituted for it in the expression cy?z, the fluent 
of the resulting quantity, being taken, will be the solidity of 
the figure proposed. | 
_ BXAMPLES. 


Exam. 1. To find the solidity of asphere, or any segment. 

‘The equation to the generating circle being y? == az — «x?, 
where a denotes the diameter, by substitution, the general 
fluxion of the solid cy?z, becomes caxz —cx*xz, the fluent of 
which gives }car*? —1ex3, or tex? (38a— 2x), for the solid 
content of the spherical segment Baz, whose height ap is zx. 

When the segment becomes equal to the whole sphere, 
then «=a, and the above expression for the solidity, becomes 
ica for the solid content of the whole sphere. 

And these deductions agree with the rules before given and 
demonstrated in the Mensuration of Solids. | 

Exam. 2. To find the solidity of a spheroid. 

Exam. 3. To find the solidity of a paraboloid. 

Exanu. 4. To find the solidity of an hyperboloid. 


eee 


TO FIND LOGARITHMS. 


108. Ir has been proved, art 23, that the fluxion of the hy- 
perbolic logarithm of a quantity, is equal to the fluxion of the 
quantity divided by the same quantity. Therefore, when any 
quantity is proposed, to find its logarithm; take the fluxion 
of that quantity, and divide it by the same quantity ; then take 
the fluent of the quotient, either in a series or otherwise, and 
it will be the logarithm sought: when corrected as usual, if 
need be ; that is, the hyperbolic logarithm. 

104. But, for any other logarithm, multiply the hyperbolic 


logarithm, above found, by the modulus of the system, for the 
logarithm sought. Note. 


'36e ‘FLUXIONS. 


‘ote. The modulus of the hyperbolic logarithms, is 1; 

and the modulus of the common logarithms, is ‘43429448190 
&c.; and, in general, the. modulus of any system, is equal to 
the logarithm of 10.in that system divided by the namber 
2-3025850929'40, &c. which is the hyp. log. of 10. Also, 
the hyp. log. of any number, is in proportion to the com. log. 
of the same number, as unity or 1 is to °43429 &c. or as the 
number 2°302585 &c. is to 1; and therefore, if the common 
log. of any number be multiplied by 2°302585 &c. it will give 
the hyp. log. of the same number ; or if the hyp. log. be di- 
vided by 2°302585 &c. or multiplied by -43429 &c. it will give 
the common logarithm. — 


Exam. 1. To find the log. of — ap 





Denoting any proposed Use z. whose logarithm is re- 
quired to be found, by the compound Scereeee eel Ne Se 


a+ 


, the fluxion of the number z, is =) and the fluxion of 


‘ata a a? a?) a4 | 
Then the fluent of these terms give the logarithm of z or 
res eG 8 x8 x4 


thm of —_ 
logar Died ~@ ~ 9a3 3a? 4a‘*t 





a : 

iting —a for ives log. Sa ee ic, 
Writing x ior x, § Corte a 20? 3a% 4aé G 
Div. these numb. and ?), . ate 2x 2a%  2n 
subtr. their logs. gives§ °° a-~ax a‘ 3a3° 5a5 


azker pee a 0-1] ata 
O e —= — {0 . —— : 
. 8 ataz 8 

















a fon 


a 
Also, because ay 


De eae ae 


Bae 7 
therefore log. of —— 1s ~ at gaa Teast aas &c 





a+z 
eet & 
and the log. of i is -+- — Seog sae TT age eo 
epee a? FER! ENC 
the prod. gives log. dee op TD as oe oe 


Now, for an example in numbers, suppose it were required - 
to compute the common logarithm of the number Oso Phe 
will be best done by the Benes) ) | 


atx 
log. Raat oh nx (+ oh eet, 


a-+x ae 
1 Mike ves a=37 ; 4 ak ne a 
Making a—x 3 g ? conseq a 3% and Va . 9? 


which 


-INFLECTIONS. 367 


which is the constant factor for every succeeding term ; also, 
Pn=2 X:43429448190 = *868588964 ; therefore the calcula- 
tion will be conveniently made, by first dividing this number 
by 3then the quotients successively by 9, and lastly these 
quotients in order by the respective numbers 1, 3, 5, 7,9, &c. 
and after that, adding all the terms together, as follows : 























3 ) -868588964 | sa 

9 ) :289529654 1) +289529654 ( °289529654 

9) 32169962 3.) 32169962 © ( 10723321 

9). 3574440 5°) 3574440 ( 714888 

Oy? SOTTO OF) 397160 ( 56737 

9°) 44129 9.) 44129" ( 4903 

9°) 4903 {| 11 ) 4903 ( 446 

9) 545| 13 ) 545 ( 42 

9.) 61| 15 ) G1 of 4 
Sum of the terms gives log. 2 = -301029995 

Exam. 2. To find the log. of sited 

Exam. 3. To find the log. of a—x. 

Exam. 4. To find the log. of 3. 

Exam. 5. To find the log. of 5. 

Exam. 6. To find the log. of 11. 


== 


TO FIND THE POINTS OF INFLEXION, OR OF CON- 
TRARY FLEXURE IN CURVES. 


105. Tue Point of EK { Pe 
Inflexion in a curve, is E TE ee ' 
that point of it which 
separates the concave 
from the convex part, | 
lying between the two; AS 
or where the curve A D A D. 
changes from concave to convex, or from convex to concave, 
on the same side of the curve. Such as the point £ in the 
annexed figures, where the former of the two is concave 
towards the axis ap, frem 4 to £, but covex from £ to F ; and 
on the contrary, the latter figure is convex from a to x, and 
concave from £ to F. | 7 

106. From the nature of curvature, as has been remarked 
before at art. 28, it is evident, that when a curve is concave 
towards an axis, then the fluxion of the ordinate decreases, 

or 


SS 


as required. 


— 368 FLUXIONS. 


or is in a decreasing ratio, with regard to, lanes of the ab- 
Sciss ; ; ‘but, on the contrary, that it increases, or is in an in- 
creasing ratio to the fluxion of the absciss, when the curve is 
convex towards the axis ; and consequently those two fluxions 
are ina constant ratio at the point of inflexion, where. the 
curve is neither convex nor concave ; that is, z ‘is to y ina 


y 


constant ratio, or + or is a constant quantity. But constant 
> x 


y 
quantities have no fluxion, or their fluxion is equal to nothing ; 
so that in this case, the fluxion of % or of = is equal to no- 


thing. And hence we have this Sener rule ; 
107. Put the given equation of the curve into fluxions ; 


from which find either 2 or ~. Then take the fluxion of this 
y 

ratio, or fraction, and ek it equal to 0 or nothing ; and from 

this last equation find also the value of the same — Rue 


; ‘ xe 
Then put this latter value equal to the former, which will 
form an equation ; from which, and the first given equation 
of the curve, x and y will be determined, being the absciss 
and ordinate answering to the point of inflexion i in the curve, 


EXAMPLES. 

Exam. 1. To find the point of inflexion in the curve whose 
equation is az? =a?y + x?y 

This ee in nase Is ane = = a8y + 2aysr + uty 





a? +4? 
= ye 
which gives ~ ry SRO hen the fluxion of this quantity 
made=0, Eres Que Se — xy) = (a? patos x (az —Ye— ary ) ; 
d thi gives? = ET y 8 


Lastly, this value of ~ being put equal the former, gives 


y 
GP fe 8 ee at ee ee OY 
a? — «x? a—y Qc ay 
or 32?==a?, and a==a,/1, the absciss. 











; and hence 2x? =a? — x2, 


ag? tas 
ata? 4g? . 
==1qa, the ordinate of the point of inflexion sought. 
Exam. 2. To find the point of inflexion in a curve defined 
by the equation ay =a ,/ ox + x, 
Exam. 3. To find the point of inflexion in a curve defied 
by the equation ay? = a’x -++ x3. 


Hence also, from the original equation, y = 


Fis sit: A. 


FLUXIONS. 369 


Exam. 4. To find the point of inflexion in, r 

the Conchoid of N icomedes, which is gene-- }—~~C 
rated or constructed in this manner: From 
a fixed point rp, which is called the pole of pes 
the conchoid, draw any number of right lines 
PA, PB, PC, PE, &c. cutting the given line rp P 
in the points F, G, H, 1, &c.: then make the 
distances, Fa, GB, HC, IE, &c. equal to each sited and equal 
to a given line ; then the éurve line ascr &c. will be the con- 
choid; a curve so called by its inventor Nicomedes. 





TO FIND THE RADIUS OF CURVATURE OF CURVES. 


108. Tue Curvature of a Circle is constant, or the same 
in every point of it, and its radius is the radius of curvature. 
But the case is different in other curves, every one of which 
has its curvature continually varying, either increasing or 
decreasing, and every point having a degree of curvature pe- 
culiar to itself; and the radius of a circle which has the same 
curvature with the curve at any given point, is the radius of 
curvature at that point; which radius it is the business of this 
chapter to find. 

109. Let are be any curve, con- ee 
cave towards its axis aD ; draw an or- 
dinate pe to the point £, where the 
curvature is to be found: and sup- 
pose Ec perpendicular to the curve, 
and equal to the radius of curvature 
sought, or equal to the radius of a 
circle having the same curvature _ 
there, and with that radius describe 
the said equally curved circle Bre ; B : RY Gg. 
lastly, draw ed parallel to an, and de 
parallel and indefinitely near to pr : thereby making ed the 
fluxion or increment of the absciss ap, also de the fluxion of 
the ordinate pe, and ze that of the curve ar. Then put z= 
AD, Y==DE, 2==a8, and r=ce the radius of curvature ; then is 
Ed=z, de=y, and Ee=z2, 

saints by sim. Loess the three lines ed, de Ee, 





or L, Y; zy 
are bosiectively. as pee saree - *\* GE, GC, CE; 
therefore - - - - GC xz =GE. ye ; 


and the flux. of this eq. is GC. 2 at GC. £=GE. yt GE. y} 
or, because cc =— 6, it isac.x2—BG. 1==GE. yTGE -y. 
But since the two curves ag and zr have the sume curva- 
ture at the point r, their abscisses and ordinates have the same 
Von It AS fluxions 


3°6 FLUXIONS. 


fluxions at that point, that is, ed. or 7 is the finxion both of ap 
and ge, and de or y is the fluxion both of pz and cz. In the 
equation above therefore substitute x for na, and y for ce. 
and it becomes 


Gtr re EFy + yy, 
or GCa—GFy =22-- y= 2". 

Now multiply the three terms of this equation respectively, 
by these three quantities, = ; a = which are all equal, and 
; Lge atone ze : 
it becomes yx—xry minny OL eG and hence is found r= 





=, for the general value of the radius of curvature, for 
Yarra e gy: 


all curves whatever, in terms of the fluxions of the absciss 
and ordinate. 

110. Furtber, as in any case either x or y may be supposed 
to flow equably, that: is, either z or y,constant quantities, or 
zor y equal to nothing, it follows that, by this supposition, 
either of the terms in the denominator, or the value of r may 
be made to vanish. Thus, when zis supposed constant; x 
being then =0, the value of ris barely - - + = + = 

z% ee vies ig. 
=. 3 or ris ==. when y1s constant. 
— zy Yr 





EXAMPLES. 


Exam. 1. To find the radius of curvature to any point of 
a parabola, whose equation is ax=y?, its vertex being a, and 
aXIs AD. 

Now, the equation to the curve being ax = y?, the fluxion 
of it is ax 2yy; and’the fluxion of this again is age = = 2y?, 
ee y constant ; hence then r or 


on SE vy 2 PAONON Coty aon ays)! or sah 42)? (a-+ 4a)? 

pa ine ~ 92 ae ae 

for the general valu of the radius of curvature af any point 
£, the ordinate to which cuts off the absciss ap=z. 

Hence, when the absciss x is nothing, the last expression 

becomes barely 1a=r. for the radius of curvature at the ver- 

tex of the parabola ; that is, the diameter of the circle of 
curvature at the vertex of a Leiba ne is equal to a, the pa- 

rameter of the axis, 

Exam. 2. To fird the: radius of curvature of an ellipse, 


seems Sr on 


whose equation is Pee -aL—x?. 
ores dng 4(a? oem x (ax — 2°).5 


Exam, 3 


FLUXIONS. | 371 


Exam. 3. To find the radius of curvature of an hyperbola, 
whose equation is a2y?==c? . ax-+-x?, 
Exam. 4. To find the radius of curvature of the cycloid, 
Ans. r==2,/aa — ax, where zx is the absciss, and a 
the diameter of the generating circle. 


OF INVOLUTE AND EVOLUTE CURVES. 


i11. An Evolute is any curve supposed to be evolved or 
opened, which having a thread wrapped close about it, fasten- 
ed at one end, and beginning to evolve or unwind the thread 
from the other end, keeping always tight stretched the part 
which is evolved or wound off: then this end of the thread 
will describe another curve, called the Involute Or, the 
same involute is described in the contrary way by wrapping 
the thread about the curve of the evolute, keeping it at the 
same time always stretched. | AL | 

112. Thus, if secu be any curve, 
and ak be either a part of the curve, . 
or aright line: then if a thread be 
fixed to the curve at H, and be 
wound or plied close to the curve, 
é&c. from H to a, keeping the thread 
always stretched tight; the other 
end of the thread will describe a 
certain curve apcp, called an Invo- 
lute; the first curve ercu being its 
evolute. Or, if the thread, fixed 
at H, be unwound from the curve, 
beginning at a, and keeping it always tight, it will describe the 
same involute agcp. 

113. If az, pF, cc, pH, &c. be any positions of the thread, 
in evolving or unwinding ; it follows, that these parts of the 
thread are always the radii of curvature, at the corresponding 
points, a, B, c,D; and also equal to the corresponding lengths 
AE, AEF, AEFG, AEFGH, of the evolute: that is, 

aE =a,eE is the radius of curvature to the point a, 

BF =aF is the radius of curvature to the point B, 

ce =a is the radius of curvature to the point c, 

DH=AH is the radius of curvature to the point p.. 


114. It also follows, from the premises, that any radius of 
curvature, BF, is perpendicular to the involute at the point.s, 
and is a tangent to the evolute curve at the point Fr. Also, that 
the evolute is the locus of the centre of curvature of the in- 
volute curve. } 





115. Hence, 


372 INVOLUTES AND EVOLUTES. 
B 








115: Hence, and from art. 109, it 
will be easy to find ome of these 
curves, when the other is given. 

To this purpose, put 


x = an, the absciss of the involute, A 
y = bb, an ordinate to the same, 
z = ps, the involute curve, 
vy = Be, the radius of curvature, 
v = EF, the absciss of the evolute, rc, 
wu = Fc, the ordinate of the same, and F 
@ = As, a certain given line. 

Then by the nature of the radius of curvature, it is 

3 
f= K —~ =Bc = aE +c; also, by sim. triangles, 
é 3 hole ay rx ez? 
Fi gst? ? GBS Heo 
. z= Ym LY 


\ cs 





ry. 





ziytir:eco= eas. 
Z yun ay 
*~ LZ Z 
Hence Er =ca—ps = ———-- —y=2. 
yu ay 
| i yz? 
and rc=ap—aE+cco=2—a-+ =; 
L- 2x 


which are the values of the absciss and ordinate of the evo- 
lute curve ec: from which therefore these may be found, 
when the involute is given. 

On the contrary, if v and wu, or the evolute be given : ‘then, 
putting the given curve ec = s, since cB=aE-+EC, or r=a-+s, 
this gives r the radius of curvature. Also, by similar trian- 
gles, there arise these proportions, viz. 





Sys Tv a-ts. | 
SaVieek i UGE, 
s 8 F _ 
eg 1 ats. 
and gi: wii nt = w= 63 
SY bhi gs 


lh 


theref. AD= ABT FC — goals ——u=f, 
& 


: | ats. 
and DB=GB—GD=GB—EF=——v —0=Y 5 
| 2 3 


which are the absciss and ordinate of the involute curve, and 
which may therefore be found, when the evolute is given. 
Where it may be noted, that s?=v?-++ 1?, and z?=x?-+- 7. 
Also, either of the quantities x, y, may be supposed to flow 
equably, in which case the respective second fluxion, x ory, 
will be nothing, and the corresponding term in the denomina- 


tor yo-ay will vanish, leaving only the other term in it; 
which 


-FLUXIONS. 373 


which will have the mgicat of rendering the whole operation 
simpler. 


116. EAE LE, 


Wns ka To determine the nature of the curve by whose 
evolution the common parabola as is described. 

Here the equation of the given involute ap, is cx = y? 
where c is the parameter of the axis ap. Hence then 


y— fet, and y = li “also jo vs by making x 


constant. Consequently the general eas of vandwu, or of 
the absciss and ordinate, Er and Fc, above given, become, i in 
that case, 








2 2 , 
EF =v eke and 
ars So ee ae 
2 
ro == ussz—a-+— — = 324+-1e—a. 


. But the value of the quantity a or az, by exam. 1 to art. 75, 
was found to be }¢ ; consequently the last quantity, rc or w Is 
barely = 32. 

Hence then, comparing the values of v and u, there is found 
3v,fe = 3us/2 or 27cv? = 16u3 ; which is the equation be- 
tween the absciss and ordinate of the evolute curve rc, show- 
ing it to. be the semicubical parabola. 

Exam. 2. To determine the evolute of the common cycloid. 

Ans.. another cycloid, equal to the former. 


TO FIND THE CENTRE OF GRAVITY. 


117. By referring to prop. 42, &c. in Mechanics, it is seen 
what are the principles and nature of the Centre of Gravity 
in-any figure, and how it is generally 
expressed. It there appears, that if 
pag bea line, or plane, drawn through 
any point, as suppose the vertex of any 
body, or figure, asp, andif - -. - 
s denote any section er of the figure, oan oe 
d = aa, its distance below pa, and B Wi 
b = the whole body or figure agp ; then 


Q 





374 CEN ERS OF GRAVITY. 


then the distance ac, of the centre of gravity below ra, ig 
sum of all the ide 











universally denoted by whether asp be a 


line, or a plane surface, or a curve Se echeici or a solid. 

But the sum of all the ds, is the same as the fluent of db, 
and } is the same as the fluent of 4; therefore tae general 
expression for the distance of the centre of gravity, is ac = 
fluent of wb fluent x6. 


“fluent of6 ~~ =~ 6 
ac. Which will divide into the following four cases. 
118. Case 1. Winer Az is some line, as a curve suppose. 


In this case § 18 = zor f/x et ys , the fluxion of the curve 
fluent of «z __ fluent of taf x? - +7? 
ze | z 
is the distance of the centre of gravity in a curve. 
119. Case 2. When the figure app is a plane; then 
b=yx; therefore the general expression becomes ac = 
- fluent of of yx 


fluent of yx 

120. Oise 3. When the figure is the superficies of a body 
generated by the rotation of a line aes, about the axis aH. 
Then, putting c = 3°14159 &c, cy will denote the circum- 
ference of the generating circle, and 2cyz the fluxion of the 
fluent of eyaz fluent of yz will 


fluent of 2cyz fluent of yz 
be the distance of the centre of gravity for a surface generat- 
ed by the rotation of a curve line z. 

121. Case 4. When the figure is a solid generated by the 
rotation of a plane ann, about the axis aH. 

Then, putting c = 3:14159 &c. it is cy? = the area of the 
circle whose. radius is y, and cyan ob the fluxion of the 
solid; therefore - - - - - = ee eye a ee 

fluent of x5 _ fluent of oye fluent of pun. beahedigs 


~ fluent of 4 ™ fluent of cy’x cyex fluent of yx 
tance of the centre of gravity below the vertex in a solid. 


; putting 2 = d the variable distance 


and 6 == z: theref. ac = 


for the distance of the centre of gravity in a plane. - 


surface ; therefore ac = 





122, EXAMPLES. _ 


Exam 1. Let the figure proposed be the isosceles triangle 
ABD. . 
a. is evident that the centre of gravity c, will be somewhere 
in the 


FLUXIONS. 375. 


in the perpendicular au. Now, if a denote 
AH, C=BD, £=aG, and y=EF ony inet 














¥ the base pp : ‘then asa:¢:: lh 
dues a 
—; therefore, by the 2d Case, sc= ots 
fluent yx 
eo arg 123 | | 
= Miami oe gat ee EA en ee a 
comes = AH: consequently cH =} an. 


In like manner, the centre of gravity of any other plane 
triangle, will be found to be at 3 of the altitude of the trian- 
gle ; the same as it was found i in prop. 43, Mechanics. 

Exam. 2. Ina parabola ; the distance from the vertex is 2x, 
er 2 of the axis. 

Exam. 3. Ina SARCaID arc; the distance from the centre 


of the circle, is 5 where a denotes the arc, c its chord, and 


x the radius. | 
Exam. 4. In acircular sector ; the distance from the centre 
: . Ser. : 
of the circle, is ‘ens where a,c, r, are the same as in exam.3. 
a 


Exam. 5, Ina circular segment; the distance from the cen- 
3 


tre of the circle is = ; were c is the chord, and a the area, 


of the segment. : 

Exam. 6. In a cone, or any other pyramid; the dinthbce 
from the vertex is 3x, or 2 of the altitude. . 

Exam. 7. In the gemisphere, orsemispheroid ; the distance 
from the centre is 3r, or 2 of the radius : and the distance 
from the vertex = of the radius. Sees 

Exam. 8. In the parabolic conoid ; the: distance from the 
base is 1a, or + of the axis. And the distance from the ver- 
tex 2 of the axis. 


Exam. 9. Inthe segment of a sphere, or of a spheroid ; the 


distance from the base is ae 





— a; where z is the height of 


the segment, and a the whole axis, or diameter of the sphere. 

Exam. 10. Inthe hyperbolic conoid ; the distance from the 
«OC 

base is - an z; where gis the height of the conoid, and « 


the whole axis or diameter. 





PRACTICAL 


Gi : { 376 | 


123. PRACTICAL QUESTIONS. 


QUESTION I. 


A rarce vessel, of 10 feet, or any other given depth, and 
ef any shape, being kept constantly full of water, by means 
of a supplying cock, at the top; itis proposed to assign the 
place where a small hole must be made in the side of it, so 
that the water may spout through it to the greatest distance 
on the plane of the base. 

Let an denote the height or side of 
-the vessel ; p the required hole in the 
side, from which the water spouts, in 
the parabolic curve pe, to the greatest 
distance Be, on the horizontal plane. 

By the scholium to prop. 68, Hy- 
draulics, the distance Be is always equal _ 
to 24/ ap . pg, which is equal to 2 4/ "x(a@—a) or 2 /aa—x?, 
if a be put to denote the whole height as of the vessel, and 


a==ap, the depth of the hole. Hence 2,/ax—x?, or az—zx?, 
must be amaximum. In fluxions, az = 2%z=0, or a—2x=0, 





and 2a = a, or x =1a. Sothat the whole p must bein the _ 


middle between the top and bottom; the same as before found 
at the end of the scholium above quoted. 


124. QUESTION If. 


If the same vessel, as in Quest. 1, stand on high, with its 
bottom a given height above a horizontal plane below ; it is 
proposed to determine where the small hole must be made so 
as to spout farthest on the said plane. . 


Let the annexed figure represent the 
vessel as before, and be the greatest 
distance spouted by the fluid, pc, on the 
plane be. 


Here, as before, ba=2 2 fad. AD . Db =2 


Jfz(e— x)= 24/ cx — x?, by putting 
ab=c, and ap==x. So that 2./ cx—2? 
or cx — x? must be a maximum. And sata like, as in the 
former question, ~ - - - - + -- ae ees 
z= ile =41ab. So that the hole p iniust be made in the 

| middle 








PRACTICAL QUESTIONS. 377 


middle between the top of the vessel, and the given plane 
that the water may spout farthest. 
¥ 
125. QUESTION Til. 

But if the same vessel, as before, stand on the top of an in- 
clined plane, making a given angle, as.suppose of 30 degrees, 
with the horizon; it is proposed to determine the place of 
the small hole, so as the water may spout the farthest on the 
said inclined plane. 

Here again (p being the place of the 
hole, and ne the given inclined plane), 
6G = 2,fad. “AD DP ae 2 f/rla—ak z), 
putting z = Bb, and, as before, a = az, 
and z==ap, Then dc must still be a 
maximum, as also Bd, being in a given G | y 
ratio to the maximum sc, on account 
of the given angle 8. Therefore ax — 

x? + xz, as well as z, is a maximum. Hence, by art. 54 of 
the Fluxions, ax— 2a, % zx = 0, ora—2rtz=0; 
conseq. + z == 2x— a; and hence be = 2 Jf 2(a — x + 2) 
becomes barely 2x. But as the given angle cpb is = 30°, 
the sine of which is 1; therefore sc = Oph or 2z, and b¢2 = 
BG* — Bb? = 32? = 3 (24—a)’, or be = + (2x <—G) 4/3. 

Putting now these two values of ba equal to each other, 
gives the equation 22 = + (2x—a) 4/3, from which is found 








_ y3 34/73 

ra re vier eee, the value of ap required. 

Note. In the Select Exercises, page 252, this answer is 
sh 


brought out —qa, by taking the velocity proporiienal to 
the root of half tthe altitude only. 


126. QUESTION IV. 





It is required to determine the size of a ball, which, being 
let fall into a conical glass full of water, shall expe! the most 
water possible from the glass ; its depth being 6, and diame- 
ter 5 inches. é 

Let azc represent the cone of the 
glass, and pune the ball, touching the 
sides in the points p and £, the centre 
of the ball being at some ‘points F in 
the axis ccefthe cone. — 


\ 





Vor. Il, 49 


378 PRACTICAL EXERCISES ON FORCES. 


Put ac =Gs = 21 =a, 
co=> 6=>06, — 
AC = o/aG?-+ac? = Gl =c 
ap = Fe = ru = or the radius of the ball. 
The two triangles acc and pcr are equiangular ; theref. 
bog 
4007 ac 3 3) DF: -8e, thatisy -@ ect’ w 2+. Fres>hence 
7 a 


Cx cx 
GF =cc— Fro =b——, and ou = or + rH = b+a——, 
a 


the height of the segment immersed in the water. Then fy 
rule 1 for the spherical segment, p. 427 vol. 1.), the content 
of the said immersed segment will be (6pr — 2cH) X GH? 


X +5236 = (2e—b +=) xX @ + b — —)? X 1-0472, 


which must be a maximum by the question; the fluxion of 
this made = O, and pac by 2x and the common factors, 














: 9 
gives ara i (fu 2) — (ae 
this reduced gives r = aoe == O41 fie radios 
Tl aaa Coan re aaa t 


of the ball. Consequently its diameter i is 411 inches, as re- 
quired. 


PRACTICAL EXERCISES CONCERNING FORCES ; 
WITH THE RELATION BETWEEN THEM AND 
THE TIME, VELOCITY, AND SPACE DESCRIBED. 


Berore entering on the following problems, it will be con- 
venient here, to lay down a synopsis of the theorems which 
express the several relations between any forces, and their 
corresponding times, velocities, and spaces, described ; which 
are all comprehended i in the followig 12 theorems, as collect- 
ed from the principles in the foregoing par ts of this work. 


Lert f, r, be any two constant accelerative forces, acting on 
any body, during the respective times ¢, T, at the end of which 
are generated the velocities v, v, and destribed the spaces, s, 
s. ‘hen, because the spaces are as the times and velocities 
conjointly, and the velocities as the forces and times ; we 
shall have, : 

1. In 


PRACTICAL EXERCISES ON FORCES. ° 379 


1. In Constant Forces, 


pet PO cr CPU Rae 
eS ory 8 gaa yeh 
ss (pte i a de ay 
Vout FT st FS 
3- fs! , FD SY no FS 
BASES sv Vi 

4, fi sche! Tis —— ied £ 
F ty 3s v2s 


And if one of the forces, as r, be the force of gravity at 
» the surface ofthe earth, and be ‘called 1, and its time r be= 
1”; then it is known by experiment that the corresponding 
space s is = 16,), feet, and consequently its velocity v==2s= 
321, which call 2¢, namely, g = 16), feet, or 193 inches. 
Then the above four theorems, in this case, become as here 
below : | 


02 
eee —= iy me ne 
5. 8 se gft igh 
ont = af ayia 

2s v $ 

76S —-— SS ~~ 
v 2Qef a 

v 8 v 
PES age gh ge 


And from these are deduced the following four theorems, 
for variable forces, viz. 


Il. In Variable Forces. 


9 8s =— Tf — are 
: {gf 
10. % = 2¢fi = ae 
an eh 
12. f =o = Y 


25 Qe¢ In 


380 PRACTICAL EXERCISES ON FORCES. 


In these last four theorems, the force f, though variable, 
is supposed to be constant for the indefinitely small time 2, 
and they are to be used in all cases of variable forces, as the 
former ones in constant forces; namely from the. circum- 
stances of the problem under consideration, an expression !s 
deduced for the value of the force f, which being substituted 
in one of these theorems, that may be proper to the case in 
hand; the equation thence resulting will determine the 
corresponding vadoes of the other quantities, required i in the 
problem. 

When a motive force happens to be concerned in the 
question, it may be proper to observe, that the motive force 
m, of a body is equal to fg, the product of the accelerative 
force, and the quantity of matter in it q; and the relation 
between these three quantities being universally expressed by 
this equation m= qf, it follows that, by means of it, any one 
of the three may be expelled out of the calculation, or else 
brought into it. 

Also, the momentum, or quantity of motion in a moving 
body, is qv, the product of the velocity and matter. 

It is also to be observed, that the theorems equally hold 
good for the destruction ef motien and velocity, by means of 
retarding forces, as for the generation of the same, by means 
of accelerating forces. 

And to the following problems, which are all resolved by 
the application of these theorems, it has been thought proper 
to subjoin their solutions, for the better information and con- 
venience of the student. 


PROBLEM I. 


To determine the time and velocity of a body descending, by the 
force of gravity, down an inclined plane; the ener of the 
plane being 20 feet, and rts height 1 foot. 


Here, by Mechanics, the force of gravity being to the force 
down the plane, : Ra length of the plane is to its height, 
therefore as 20: 1 (the force of gravity) : 3; =/f the 
force on the aia 

Therefore, by theor. 6, v or ,/4gfsis \/ 4 X 163, X54 X 
20=./4% 161,==2 X4,; or 8), feet nearly, the last “velocity 
per second. And, 

20 400 0 
Rae — 376 
By theor. 7, ¢ or ./ PV ie LX =V ie i; ae = 318 
seconds, the time of descending. 
PROBLEM 


PRACTICAL EXERCISES ON FORCES. 381 


PROBLEM II. 


If a cannon ball be fired with a velocity of 1000 feet per second 
up a smooth inclined plane, which rises 1 foot in 20: it is 
proposed to assign the length which it will ascend up the plane, 
before it stops and begins to return down again, and the time 
of ite ascent. 


Here f=; as before. 


1 tle 10007 600006006 
. : Sa te i 
hen, by theor. 5, s Rat AGA 38 
= 310880182 feet, or nearly 59 miles, the distance moved. 
v 1000 120000 
And, by theor. 7, = — = ——-______ = = 
aka GaP MORAG RN FOS 


621147 = 10’ 217147, the time of ascent. 


PROBLEM III. 


If a ball be projected up a smooth inclined plane, which rises 1 
foot in 10, and ascend 100 feet before it stop: required the 
time of ascent, and the velocity of projection. 


First, by theor. 6, v = 4/ 4g fs = 4/4 X 16,5 Xj), X100= 
ziss/ 10=25-36408 feet per second, the velocity. 


s 100 10 
And, by theor. 7; t EM ay VM ipdy gag SO = 13 
4/ 10=17-88516 seconds, the time in motion, 





PROBLEM IV. 


If a ball be observed to ascend up a smooth inclined plane 100 
feet in 10 seconds, before it stop, to return back again: re- 


guired the velocity ef projection, and the angle of the plane’s 
inclination. 


Qs 200 
_ First, by theor. 6, v == aie = 20 feet per second, 
the velocity. | 
8 100 12 


And, by theor. 8, f= —--=---____-=—-. Thatis, th 
nd, by theor 8, f gi 162. x 100 193 at is, the 


length of the plane is to its height, as 193 to 12. 

Therefore 193 : 12: : 100: 6-2176 the height of the plane, 
or the sine of elevation to radius 100, which answers to 3° 34’, 
the angle of elevation of the plane. 

PROBLEM 


382 PRACTICAL EXERCISES ON FORCES. 


PROBLEM. V. 


By a mean of several experiments, I have found, that a cast cron 
ball, of 2 inches diameter, fired perpendicularly tnto the face 
or end of a block of elm wood, or in the direction of the fibres, 
with a velocity of 1500 feet per second, penetrated 15 inches 

deep into its substance. It is proposed then to determine the 
time of the penetration, and the resisteng force of the wood, 
as compared to the force of gravity, supposing that force to 
be a constant quantity. 


28 2X13 


1 
First. by theor. 7, ¢ = leno ie 658 part of ase- 


second, the time in penetrating. 


y2 1500? 81000000 

Bnd, by ther bie aa a oe Ma 18 92 doe 

= 32284. Thatis, the resisting force of the wood, is tothe 
force of gravity, as 32284 to 1. | 

But this number will be different, according to the diame- 


ter of the ball, and its density or specific gravity. For, since 
2 
fisas —by theor. 4, the density and size of the ball remain- 


ing the same ; if the density, or specific gravity, , vary, and 
all the rest be constant, it is evident that f will be as n ; and 


2 
therefore f as = when the size of the ball only is constant. 


But when only the diameter d varies, all the rest being con- 
stant, the force of the blow will vary as 4? or as the magnitude 


of the ball ; and the resisting surface, or force of resistance, 
3 


verge 
varies as d? ; there/f is as qa OF as d tees when all the rest 


‘ d 
are constant. Consequently fis as a when they are all 


variable. 
dnv? dno? 
And oe so and ; where f denote the 
F  BNV?s s pnv2f 
strength or firmness of the substance penetrated, and is here 
supposed to be the same, for all balls and velocities, in the 
same substance, which is either accurately or nearly so. See 
page 581, &c. vol. 1, of my Tracts. 
Hence, taking the numbers in the problem, itis - - - 
dnv? 2, X 72. X 18002 _ 44X 15002 
f=—=> = ee = —-—+—_——_ = 2538462 the 
§ Te. 39 
value of f for elm wood. Where the specific gravity of | 
© the 





PRACTICAL EXERCISES ON FORCES. 383 


the ball is taken 71, which is‘a little less than that of solid 
cast iron, as it ought, on account of the air bubble which is 
found in all cast balls. 


PROBLEM VI. 


To find how far a 2416. ball of cast iron will penetrate into a 
block of sound elm, when fired with a velocity of 1600 feet 
per second. a 


Here, because the substance is the same as in the last 
problem, both of the balls and wood n = n, and r=f; there- 


2 DVv?s__ 5°85 X 1600? X13 


s V : 
——=—___ SS SO 1-3. 
fore -=——, ors nee 2315002 41.2. inchés 


D 
s - dv 
nearly, the penetration required. 


¢ 


PROBLEM VII. 


It was found by Mr. Robins, (vol. 1. p. 273, of his works), 
that an 18-pounder ball, fired with a velocity of 1200 feet 
per second, penetrated 34 inches into sound dry oak. It is re- 


quired then to ascertain the comparative strength or firmness 
of oak and elm. 


Tue diameter of a 16lb ball is 5:04 inches =p. Then, 
by the numbers given in this problem for oak, and in Prob. 5, 
forelm,we have - - - - - s pele se 





f_dv?s 2X 1500? X34: 100X17—__:1700 ; 
F pv2s 504% 1200? X13 504x16X13 1048 “ ~ * 
nearly. 


From which it would seem, that elm timber resists more 
than oak, in the ratio of about 8 to 5; which is not probable, 
as oak is a much firmer and harder wood. But itis to be 
suspected that the great penetration in Mr. R’s experiment 
was owing to the splitting of his timber in some degree. 


PROBLEM VII. 


A 24-pounder ball heing fired into a bank of jirm earth, with « 
velocity of 1300 feet per second, penetrated 15 feet. It is 
required then to ascertain the comparative resistance of eli 
and earth. 


Companine the numbers here with those in prob. 5, if 


4S 


POOP Cage gl 
ae te 


384 PRACTICAL EXERCISES ON FORCES. 


1800 — 20 . 
3 








A a dv?s 2K 1500? X15 bee 15? X 24 
F pv2s 5-65x13002X13  13°x0-37 27? 
nearly = 6 3 nearly. That is, elm timber resists about 62 


times more hen earth. 


PROBLEM IX. 


To determine how Fre a leaden bullet, of 3 of an inch diameter, 
will penetrate dry elm; supposing it fired with a velocity of 
1700 feet per second, and that the lead does not change tts 
figure by the stroke against the wood. 


Here p=2,n=11l1,n=71. Then bY: the lakh and 
theorem in prob. 5, itiss = - 


DNV73s_~ 3X 111X17002 X13 173 X13 63869 = 92 ‘aches 
pt a Sa cn er ets Sat in 
dnv2 ERTS X 15002 200 X33 6600 


nearly, the depth of penetration. 


But as Mr. Robins found this penetration, by experiment, 
to be only 5 inches; it follows either that his timber must 
have resisted about twice as much; or else, which is much 
more probable, that the defect in his penetration arose from 
the change of figure in the leaden ball he used, from the blow 
against the wood. . 


PROBLEM X. 


A one pound ball, projected with a velocity of 1500 feet per 
second, having been found to penetrate 13 inches deep into 
dry ine It 2s required to ascertain the time of passing 
through every single inch of the 13, and the velocity lost at 
each of them; supposing the resistance of the wood constant 
or uniform. 


Tue velocity v being 1500 feet, or 1500 X 12 = 18000 
inches, and velocities and times being as the roots of the 
spaces, In constant retarding forces, as well as in accelerating 

VeRG nears 13 1 
12 1500 9000 692 
second, the whole time of passing through the 13 inches ; ; 
theféfore, as 

a/ 13 


: 2 
ones, and ¢ being Be = part of a 


PRACTICAL EXERCISES ON FORCES.’ 








: 13—./12 wee Gees 


385). 


£ JIS 
J veloc. lost Time in the 
as 3 
er ae BS9o3s ts Vidi, “00005 Istinch. 
12—,/11 we 11 
13 /13 
PItH 4710 Sli S10, 
= D& 
713 64:2 &c Vi3 = -00006 3d 
10—,/9 10— 
Jf 10- J: » = 675 fills salt —-——-¢ == 00007. 4th _ 
13 ae ae ; 
VSPA aay BC Ades, Abe an 00007 5th 
/13 | ia 


a » = 81-7 
i oa = 88-8 
Jo-Jf4 








See aro) 
1—,/0 
ae » = 416:0 
Sum 1500-0 





= -00007 6th 


| vine —t = -00008"7th 


GREASE 
Soe = -90008 8th 


Pads = -90009 9th. 





Sum <1; or °00144 sec. 


692 





Pee = -00011 10th 
Sanatan i=. 00013 11th 
ae = 00017 12th 


v Saat ae a 00040 13th 


Hence, as. . the motion. lost at the bene a is very small ; 
and consequently the motion communicated to any body, as __ 
an inch plank, in passing through it, is very small-also; we 
can conceive how-such a plank | may be shot nrOUED when 


standing HPHEYS without oversetting it. 
I 50 


Vou. 


PROBLEM 


386 «= PRACTICAL: EXERCISES ON FORCES. 


- PROBLEM XI. 


The force of attraction, above the earth, being inversely as the 
square of the distance from the centre ; it is proposed to deter- 
mine the time, velocity, and. other circumstances, attending @ 
heavy body falling Peet any given height ; the descent at “the 
earth's surface being 16}, feet, or 193 inches, in the first se- 
cond of time. 


Put 


y == cs the radius of the earth, 

a= ca the dist. fallen from, 

ax == cP any variable distance, 

v = the velocity at Pp, 

t = time of falling there, and 

g = 16,1, half the veloc. or force at s, 
Jf == the force at the point r. 





Then we have the three following equations, viz. 
2 ; 
EES Saat ee oe ie ~ the force at p, when the force of gravity is 


considered as he ty = — x, because a“ decreases ; and 


ep 
ve — Rafe — OE, 


io ates Agr? 
The fluents of the last equation give v2 se But 


when x ==a, the velocity y=> 0; therefore, ty correction, 
a ue a a Agr? x —; orv=4/ (-— si x=), 
a general expression for the saincity at any ink Pe 

When « = r, this gives 0 = of (Agr X —") for the 


greatest dchocliy, or the velocity when a mcs ities the 
earth. 


02 = 





‘When a is Nery great 1 in respect of r, the last velocity be- 
comes (1 — =) xX. /4er very nearly, or nearly 4/ 4gr only, 


which is: actiwately the greatest velocity by falling from an 
infinite height. And this, when vr — 3965 miles, is 6.9506 


miles per second. Also, the relsoug acquired in falling from. 
the 


PRACTICAL EXERCISES ON FORCES, 387 


the distance of the sun, or 12000 diameters of the earth, in 
6°9505 miles per second. And the velocity acquired in fall. 
ing from the distance of the moon, or 30 diameters, is 6:8972 
miles per second. 

Again, to find the time; since ty = = x, therefore = 
—% a — LE 








= the correct fluent of which gives 
eed Agr? 7 7 Pinar LI % 
t= aes (/ ar—za-+ arc to en a. sen vers. a—2); 


or the time of falling to any point p = ooge =v X (4B 4. BP). 


_ sks 


And when x = f, this becomes t = by x — for the 


whole time of falling to the surface at s ; which i is evidently 
infinite when a or acis infinite, though the velocity is then 
only the finite quantity /4gr. 

When the height above the earth’s surface is given =g ; 
because ris then nearly =a, and ap aD. nearly = = ps, the time ¢ 
for ne distance g willbe nearly - - - + .- - + - » 


J Ave aw a : Xf 4g i, as it ought to be. 
r=l1, 
gre AT 


Ifa a at the distance of the moon at a, fall to the earth’s 
surface ats. Then r=3965 miles, a=60r, and t = 416806” 
=4 da. 19h. 46’ 46”, which is the time of falling from the 
moon te the earth. 

‘When the attracting body is considered as a point c; the 














wale time of descending toc willbe - - - + - =°% 
”, -7854a a 10a - ai 7854 at 
avs ae g bir’ r 


Henke the times employed by bodies, in falling from qui- 
escence to the centre of attraction, are as the square roois of 
the cubes of the heights from which they respectively fall. 


PROBLEM XII. « 


The force of attraction below the earth’s sunfiice bese directly 
as the distance from the centre; wt is proposed to determine 
the circumstances of velocity, time, and space fallen by a heavy 
body from the surface, through o perforation made straight to 
the centre of the earth: abstracting from the effect of the earth’s 
rotation, and ei aii at to be w | homogeneous sphere of 3965 
miles radius. 


Put 


~~ 


388 PRACTICAL EXERCISES ON FORCES. - 


Put + = ac the radius of the earth, 
. o = cp the dist. from the centre, 
0 = the velocity at Pp, 
-t = the time there, 
= 16,5, half the force at as 
= the force at P. 
(Then ca: cp:: 1 : f; and the three 





: Miele aR: 
equations are rf = «, and vwy= — 2g fe, and v7 =—z- 


Hence f = “, and vy = <2 = ; the correct fluent of which 





Shes 242 
gives v=4/ (2g x )= pos = Poy, the velocity: at 


the point p; where pp and ce are perpendicular toca. Se 
that the velocity at any point P, is as ms perpendicular or 
sine pp at that point. 


“When the body arrives at c, thenv = J r= J 2g . AC 


-==25950 feet or 49148 miles per second, which is the great- 


est velocity, or “that at the ‘centre c. 
Again, for the time; z= ek Artie hee 
z 28 fr? — a2 





; and the 





lee 022 Ok au Nes 1 
fluents give t= VigzX% arc to cosine Vag x arc ap. So 


that the time of descent to any point P, is as the correspond- 


ing arc AD. 
ee P arrives at.c, the above iecnmes ee ERE ares se ae 
Vx _% quadrant Ak == cas = 1: 5108 / a = 12673 se- 


cae = 21'7’1, for the time of falling to the ee c. 

The time of falling to the centre is the same quantity 1°5708 
J = from whatever point in the radius ac the body begins 
tomove. For, let x be any given distance eae cat which 


the motion commences : tei by: correction, = Ge ‘ue 














and hence 7 =,/— x. 
V5 a leas 

ae x arc to cosine ~ me meh when; ag = 0, gives t= / s- & 

quadrant = 1: 5708 V5 — ” for the time of descent to the cen- 


tre c, the same as batons. 
As 


PRACTICAL EXERCISES ON FORCES. 39. 


As an equal force, acting in contrary direction, generates 
or destroys an equal quantity of motion, in the same time ; 
it follows that, after passing the centre, the body will just 
ascend to the opposite surface at B, in the same time in which 
it fell to the centre froma. Then from it will return again 
in the same manner, through c to 4; and so oscillate continu- 
ally between a and s, the velocity being always equal at equal 
distances from ¢ on both sides; and the whole time of a dou- 
ble oscillation, or of -passing from a and arriving at a again, 
will be quadruple the time of passing over the radius ac, or 


=2X31416/5- = th, 24/ 29", 


PROBLEM XII. 


To find the Time of a Pendulum vibrating in the Are of @ 
Cycloid. ue 


Let s be the point of suspen- 
sion; sa, the length of pendu- 
lum ; cas, the whole cycloidal 
arc; alkp, the generating cir- 
cle, to which FKE, HIG are per- 
pendiculars. 

sc, sB two other equal semi- 
cloids, on which the thread 
wrapping, the end « is made to” 
describe the cycloid Bac. — | prste. 
- Now, by the nature of the cycloid, an =ps; and sa = 
Qap = sc = sB = sa = as. Also, if at any point c¢ be 
drawn the tangent cp; also ce parallel and pq perpendicular 
to ap. Then pe is parallel to the chord ar by the nature of 
the curve. And, by the nature of forces, the force of gravity : 
force in direction @P::Ge:GqQ:: al: AH :: AD : ar; in 
like manner, the force of gravity: force in the curve ate: : 
ap : aK; that is, the accelerative force in the curve, is every 
where as the corresponding chord a1 or 4x of the circle, or as 
the arc ac or ax of the cycloid, since ac is always = 2ar, by 
the nature of the curve. So that the process and conclusions, 
for the velocity and time of describing any arc in this case, 
will be the very same as in the last problem, the nature of 
the forces being the same, viz. as the distance to be passed 
over to the lowest point a. 7 





From 


390 «© PRACTICAL EXERCISES ON FORCES. 


From which it follows, that the fine of a. ieetcan Set 
in all aye AG, AE, &c. is ha same constant quantity Bei 


l 
i: BOB 5-= 1 B18 = elt "BT0BY/ 5 and the time ofa 


whole vibration from B to Cs or from c to 8, is 3° inet, 


where L=as=as is the length of the pendulum, g = ie 
feet or 193 inches, and 3°1416 the circumference of a circle 
whose diameter 1 1s 1. ; 


Since the time of a body’s falling by gravity through 47, 
or half the length of the pendulum, by the nature of descents, 


i 
is 4/5 which being in proportion to 31416 gos as 1 js to 


8:1416; therefore the diameter of a circle, is to its circum- 
ference, as the time of falling through half the length of a 
pendulum, is to the time of one vibration. 


If the time of the whole ater be 1 second, this equa- 


28 g 
PPS iis Tie pe ee i: OP a 

tion arises, viz. 1"=3: 1416/5 ; hence l eT T MEY Vs 
and g = 3°14162 X11=— 4: aes? So that if one of these, 
g or l, be given by experiment, these equations will give the 


other. When g, for instance, is supposed to be given =16,), 


eeu Lae 
feet, or 193 inches ; then is J oe 39°11, the length of 


a pendulum to vibrate seconds. Or if 1== 391, the length of 
the seconds pendulum for the latitude of London, by experi- 
ment; thenis g = 4:93481 = 193-07 inches = 16,1,%7 feet, or 
nearly 16,1, feet, for the space descended by gravity in the 
first second of time, in the latitude of London ; also agreeing 
with experiment. 


Hence the times of vibration of pendulums, are as the 
square roots of their lengths ; and the number of vibrations 
made in a given time, is reciprocally as the square roots of 
the lengths. And hence also,. the length of a pendulum 
vibrating | n times in a pupae or Rose is a 393 Xx ‘ae 


60 140850. 
n nN 


When a pendulum vibratesin a circular arc ; as the length 
of the string is constantly the same, the time of vibration 
will be longer than in a cycloid; but the two times will ap- 
proach nearer together as the circular arc is smaller ; a ia 

1eL 


PRACTICAL EXERCISES ON FORCES. 391 


when it is very small} the times of vibration will be nearly 
equal. And hence it happens that 391 inches is the length 
of a pendulum vibrating seconds, in the very small are of a 
circle. iy . 


PROBLEM XIV. 


To determine the Time of a Body descending down the Chord of 


a Circle. 


Lert ec be the centre ; as the vertical 
diameter ; ap any chord, down which a 
body is to descend from P to a; and ra 
perpendicular to aB. — | 

Now, as the natural force of gravity in 
the verticai direction Ba, is to the force 
urging the body down the plane pa, as 
the length of the plane ap, is to its height 
Aq ; therefore the velocity in pa and Qa, 
will be equal at all equal perpendicular 
distances below ra; and consequently the 
time in Pa : time in @A :: PAC? Qa tf Bal Re: Dat 
time in BA: time INQA :: 4/BA : 4/QA :: BA! Pa; 
hence, as three of the terms in each proportion are the same, 
the fourth terms must be equal, namely the time in sa = the 
time Pa. : : : 

And, in like manner, the time in gp = the time in Ba. So 
that, in general, the times of descending down all the chords 
BA, BP, BR, BS, &c. or PA, RA, Sa, &c. are all equal, and each 
equal to the time of falling freely through the diameter ; as 





before found at art. 131, Mechanics. Which time 1s ces é 
where g = 16,1, feet, and r =the radius ac ; > - - - 
7 2r : 
for /gi/2ri:t as ee 
PROBLEM XV. 


To determine the Time of filling the Ditches of a Work with 

- Water, at the Top, by a Sluice of 2 Feet square; the Head 
of Water above the Sluice being 10 Feet, and the Dimensions 
of the Ditch being 20 Feet wide at bottom, 22 at Top, 9 deep, 
and 1000 Feet long. 


‘Tne capacity of the ditch is 189000 cubic feet. 
But ./ g : 4/ 10:: 2g : 2,710g the velocity of the water 
through the sluice, the area of which is 4 square feet: 
therefore 


392 PRACTICAL EXERCISES oN ‘FORCES. 


therefore 8 Vv 10g is the quantity. per second: running through 
sana . 
1b}; and consequently 8y/10g : - 189000 : 1’; = 


or 31! 3 nearly, which is the time of ing the ditch. 
PROBLEM xVL. 


To determine the Time of emptying a : Vausel of Water by a 
Sluice in the Bottom of wt, or in the Side near the Bottom : the 
Height of the Aperture being very small in ee of .the Al- 
titude of the Fluid. . | 


Pur a = the area of the ipertine! or sluice; — 
2g == 321 feet, the force of gravity ; ; 
d= the whole depth of water ; 
« =the variable altitude of the surface above the 
aperture ; 
A = the area of the surface of the water. 


Then / g gs: fies Beis 2Q./en the velocity with which the 
fluid will issue at the sluice; and hence 4 : a: : Q/gat cewlee 


the velocity with which the surface of the water will deckene 
at the altitude x, or the space it would descend in 1 second 
with the velocity there. Now, in descending the space x the 
velocity may be considered as Bogen and uniform descents 
at Sap wii I’: aed the 
2a,/gu . 

time of descending « space, or the fluxion of the time of ex- 








areas their times ; therefore 


hausting. _ _ That is, 7 = ery a3 which is. made negative, be- | 


cause 2 is a decreasing quanty. or its flaxion peers 


Now, when the nature or figure of the yest is given, the 
- area a will be given in terms of 2; which value of a being 
substituted into this fluxion of the time, the fluent of the result 
will be the time of exhausting sought. 


‘So if, for example, the vessel be any prism, or every 
where of the same breadth ; sg Aisa constant quantity, - 


and therefore the fluent is — — a5 *. But when 2 = d, this 
ood 
gee af 7 and should be 0; alierstore the correct 


Auent i ist=—~ = X ws for the time of the surface. Je: 


¢ 


scending 


PRACTICAL EXERCISES ON FORCES. 393 


gcending till the depth of the water be x.. And when x=0, 
} Fay d 
the whole time of exhausting is barely == 


Hence, if a be = 10000 ‘square feet, a == ! square foot, and 
d == 10 feet; the time is 78851 seconds, or 2h 11’ 25"1. 
Again, if the vessel be a ditch, or canal, of 20 feet broad 
at the bottom, 22 at the top, 9 deep, and 1000 feet long ; 
then is 90: 90 +x 1 203 ee % 2 the breadth of the 
surface of the water when its depth in the canal is x; and 


therefore pat x 2000 is the surface at that time, con- 


sequently ¢ or “7 _=1100 x tal x ——_ is the flur- 
2A gu 9 af ZX 
ion of the time ; the correct fluent of which, when x = 0,13 
180+2d d_ 1000X186x3 _ o 
1006. K berg eens aya 15459”2, nearly, 
or 4h. 17 39/2, being the whole time of exhausting by a sluice 
of 1 foot square. 











PROBLEM XVII. 


To determine the Velocity with which a Ball ts discharged from 
a Given Piece of Ordnance, with a Given Charge of Gun- 
_ powder. 





Ler the annexed figure, 
represent the bore of the _B C £ 
gun; ap being the part |e © 
filled with gunpowder. D 
And put 


a = as, the part at first filled with powder and the bag; 

6b = az, the whole length of the gunbore ; 

c = +7854, the area of a circle whose diameter is 1; 

d = gp, the diameter of the ball: ! 

e = the specific gravity of the ball, or weight of 1 cubic foot ; 

g = 16,5 feet, descended by a body in 1 second ; 

m= 230 ounces, the pressure of the atmosphere ona sq. inch ; 

n to 1 the ratio of the first ferce of the fired powder, to the 
pressure of the atmosphere ; 

w= the weight of the ball. Also, let 

x == ac, be any variable distance of the ball from a, in moving 
along the gun-barrel. 

Vor. Il. 51 First, 


$9 PRACTICAL EXERCISES ON FORCES: 


. First, cd? is = the area of the circle sp of the ball ; 
there mcd? is the pressure of the atmosphere on Bp ;__ 
censeq. mned? is the first force of the powder on Bp. 

But the force of the inflamed powder is proportional to its 
density, and the density is inversely as the space it fills ;- 
therefore the force of the powder on the ball ats, is to the 
force on the same atc, as Ac isto ap; thatis,- - - -. + 


mnacd? 


22a ::mncd2 ; me Meare the motive force at c: 





F mnacd? 
conseq. met , Le cee force there. 
: 2 d2 x 
Hence, theor. 10 of forces gives vy = 22fx= —_ x= 5 








the fluent of which is 0? = X hyp. log. of x. 


4gmnacd? 
w 
- But when v=0, then x=a ; theref, by correction, 
4gmnasd? 
w 
sa 


ots * hyp. log. = is the correct fluent : conseq: 


X hyp. log. =) is the vel. of the balf at c. 
(Canned 


tage operons 
and v= 4/ (—— 


the ball issues eens the muzzle at x; where h denotes the 
length of the cylinder filled with powder; and a the length to 
the hinder part of the ball, which will be more than h wher 
the powder does not touch ‘the ball. 


* hyp. log. * the velocity with which 


Or, by substituting the numbers for g and m, and chang- 
ing the hyperbolic peti for the common ones, then 
yf Gees X com. log. ey, the velocity at x, in fect. 

But, thecontent of the balk being 2cd°, its weight i is = - 
n= SE gs aaao which being substituted for w, in 

the value of v, it becomes 
= oT13y/ (5X com. log. .the velocity ate. 
When the he is of cae iron; taking e=7368,the rule becomes: 
9=100,/(— 10 x log. 2) for the veloc, of the’ ‘cast-iron ball. 

Or, when the ball i z af dead: thane (+ \4:. “ve wae 

| a= 208 (a ra aX log, for the veloc. of the leaden ball, 


Corol. 


PRACTICAL EXERCISES ON FORCES, 895 


Corol.. Fromthe general expression for the velocity v, above 
given, may be derived what must be the length of the charge 
of powder a, in the gun-barrel, so.as to produce the greatest 
possible velocity in the ball; namely, by making the value of 
v a Maximum, or, by squaring and omitting the constant quan- 


tities, the expression aX hyp. log. of a2 maximum, or its flux- 
lon ia te nothing ; that is ax hyp. log. ee == 0, or hyp. 
tog. of = -= 1, hence == 271828, the number whose hyp. 


log. is i. So that a: 6: : 1: 2°71828, or as 4 to 11 nearly, 
or nearer as 7 to 19; that is, the depeth of the charge, to pro- 
duce the greatest velocity, is the 4th part’ at the length of 
the bore, or nearly ¥, of it. 


But by actual experiment it is found, that the charge for the 
greatest velocity, is but little less than that which is here 
computed from-theory ; as may be seen by turning to page 
252 of my volume of ‘Tracts, where the porresvontis Pe 
are found to be, for four different lengths of gun, thus ¥;, 7%, 
vss 33 the parts here varying, as the gun is longer, which 
allows time for the greater quantity of powder to be fired, 
before the ball isout of the bore. 


SCHOLIUM. 


In the calculation of the foregoing problem, the value of 
the constant quantity » remains to be determined. It denotes 
the first strength or force of the fired gunpowder, just before | 
the ball is moved out of its place. This value is assumed, by 
Mr. Robins, equal to 1000, that is, 1000 times the pressure 
of the atmosphere, on any ‘equal spaces. — 

But the value of the quantity » may be derived much more 
accurately, from the experiments related in my Tracts, by 
comparing the velocities there found by experiment, with the 
rule for the value of ee or the ae as above a i Py 
theory,viz. v=1004/(— 0 => xlog. fe “)? or=100y/(— 7 7 xlog: tof =). 
Now, supposing that » is a given igang as well as all sig 
other quantities, excepting only the number, z, then by re-. 
ducing this equation, the value of the ae n 1s found to be as 


‘ & 
follows, viz. n=———__ oee — com. log. of 3 or= = log. of. 


1000a 
when h is different rhe a. 


dvv 
100k * 


396. PRACTICAL EXERCISES ON FORCES. 


‘Now, to apply this to the experiments. By page 240 of 
the Tracts, the velocity of the ball of 1:96 inches diameter, 
with 4 ounces of powder, in the gun No. 1, was 1100 feet per 
second; and, by pa. 494, vol. 1, the length of the gun, when 
corrected for the spheroidal hollow in the bottom of the bore, 
was 28°53; also, by page 228, the length of the charge, when 
corrected in like manner, was 3°45 inches of powder and bag 
together, but 2°54 of powder only: so that the values of the 
quantities in the rule, are thus : a=3°45; b=28-53 ; d=1°96 ; 
h=2-54; andv = 1100: then, by substituting these values 


’ ; d b 
instead of the letters, in the theorem qe ee com. log. of —, 
1000a a 


it comes out n = 750, when h is considered as the sameas a. 
And so on, for the other experiments there treated of. 


It is here to be noted however, that there is a circumstance 
in the experiments delivered in the Tracts, just mentioned, 
which will alter the value of the letter a in this theorem, 
which is this, viz. that a denotes the distance of the shot from 
the bottom of the bore ; and the length of the charge of pow- 
der alone ought to be the same thing ; but, in the experiments, 
that length included, besides the length of real powder, the 
substance ef the thin flannel bag in which it was always con- . 
tained, of which the neck at least extended a considerable 
length, being the part where the open end was wrapped and 
tied close round with athread. This circumstance causes 
the value of m,.as found by the theorem above, to come out 
less than it ought to be, for it shows the strength of the in- 
flamed powder when just fired, and when the flame fills the 
whole space a before occupied both by the real powder and 
the bag, whereas it ought to show the first strength of the 
flame when it is supposed to be contained in the space only 
occupied by the powder alone, without the bag. The formula 
will therefore bring out the value of m too little, in proportion 
as the real space filled by the powder is less. than the space 
filled both by the powder and its bag. In the same propor- 
tion therefore must we increase the formula, that is, in the 
proportion of h, the length of real powder, to a the length of 
powder and bag together. When the theorem is so correct- 
ed, itb ae log of = 
e ri ecomes Toooh = com. log. of -. 

Now, by pa. 228 of the Tracts, there are given both the 
lengths of all the charges, or values of a, including the bag, 
and also the length of the neek and bottom of the bag, which 
ts 0:91 of an inch, which therefore must be substracted from 

| all 


PRACTICAL EXERCISES ON FORCES. 397 


all the values of a, to give the corresponding values of h. 
This in the example above reduces 3°45 to 2.54. 

Hence, by increasing the above result 750, in proportion 
of 2°54 to 3.45, it becomes 1018. And so on for the other 
experiments. a 

But it will be best to arrange the results in a table, with 
the several dimensions, when corrected, from which they are 
computed, as here below. , 


Table of Velocities of Balls and First Force of Powder, &c. 




















Gun. Charge of Powder. 
$$$ | —___ __--—__—__| Velocity; First 
Length, | Weight | Length or | or value force, or 
No.| or value in value of v. |value of | 
of 0. ounces. lof a. | of kh. : n. 
Leones Sg By re ee ws A Yicce008 BRITO ARNE SY EEA SESE 
inches. 4 3°45} 2°54 1100 1018 | 
1] 28-53 8 | 5:99] 5-08] 1430 1164 | 
16 {11°07/10°16}) 1430 967 
4 | 3-45) 2-54} 1180 | 1077 | 
sae 38°43 8 5°99} 5:08 1580 1193. 4 
16° (11°07}10-16 1660 © 984 





| 4 | 3°45) 2541 1300 | 1067 

3 | 57°70 8 | 5-991 5:08] . 1790 1256 

| 16 |11-07110-16! 2000 1076 
HO BA We OY, GSS carers Sis a 


| B45] 2°54 





4 1370 1060 
| 4 80-23 8 | 5-99 508 | 1940 1289 
| 16 111-07110-16 


2200 | 1085 





Where it may be observed, that the numbers in the column 
of velocities, 1430 and 2200, are a little increased, as, from a 
view of the table of experiments, they evidently required to 
be. Also the value of the letter d is constantly 1:96 inch. 

Hence it appears, that the value of the letter n, used in 
the theorem, though not yet greatly different from the num- 
ber 1000, assumed by Mr. Robins, is rather various, both for 
the different lengths of the gun, and for the different charges 
with the same gun. 


But 


398 PRACTICAL EXERCISES ON FORCES. 


But this diversity in the value of the quantity a, or the first 
_ force of the inflamed gunpowder, is probably owing in some 
measure to the omission of a material datum in the calculation 
of the preblem, namely, the weight of the charge of powder, 
which has not all been brought into the computation. For it 
is manifest, that the elastic fluid has not only the ball to move 
and impel before it, but.its own weight of matter also. The 
computation may therefore be renewed, in the ensuing pra- 
blem, to take that datum into the account. 


PROBLEM XVIII. 


Zo deéermine the same as in the last Problem; taking both the 
Weight of Pewder and the Ball inte the Calculation. ~ 


Besrpes the notation used in the last problem, let 2p denote 
the weight of the powder in the charge, with the flannel bag 
im which it was inclosed. . 

Now, because the inflamed powder occupies at all times the 
part of the gun bord which is behind the ball, its centre of 
gravity, or the middie part of the same, will move with only 
half the velocity that the ball moves with ; and this will require 
the same force as half the weight of the powder, &c. moved 
with the whole velocity of the ball. Therefore, in the con- 
clusion derived in the last problem, we are now, instead of w, 
to substitute the quantity p + w; and when that is done the 

ashen , 2230nhd?2 
last velocity will come onto Ca 





b 
X com. log. ma, 


And from this equation is found the value of », which is 
Bik Bas PO Ctl Pah en ee at 
5v? = log. Of a argent > log. of-, by sub 





& 
3: 


New as to the ball, its medium weight was 16 oz. 13 dr. == 
16°81 oz... And the weights of the bags containing the several 
charges of powder, viz. 4 0z. 8 oz. 15 oz. were 8 dr. 12 dr, 
and | oz. 5 dr; then adding these to the respective contained 
weights of powder, the sums, 4:5 oz. 8°75 0z. 17°31 oz. are 
the values of 2p, or the weights of the powder and bags ; 
the halves of which, or 2°25, and 4°38, and 8:66, are the 
_ values of the quantity p for those three charges ; and these 
being added to 16°81, the constant weight of the ball, there’ 
are obtained the three values of p + w for the three charges 
of powder, which values therefore are 10-06 oz. and 21:19 
oz. and 25°47 6z. Then, by calculating the yalues of the 
first force n, by the last rule above, with these new data, the 
whole will be found as in the following table. | 

The 






e gun. Charge of Powder. : Weight of First | 


Velocity, 







































































-- ball and or the force! 

Length Weight Length or |charge,or| orthe; 

» values 

iNo. or value}; in value values of | ofp, (value 
of 6. |ounces.jof a. lof hk. | po - of n | 
—|——$—— : CS EE Seay ee ee | 
inches. 4 3°45) 2°54 1906 } 1100 | 1155, 
| bP} 98-53 8 | 5°99! 5-08] 21-19 | 1430° | 1476 
16 }12-07/10°16) 25°47 1430. | 145%) 
pores. Ws : \ 
4 >| 3-45) 2-54] 19-06 | 1180) | 1167) 

2 | 38°43 8 5°99| 5°08) 21-19 1580 | 1506; 
16 {11-O7|10°16| 25°47 1660 | 1492) 
ates at a Tie lla i mE SIE RBA sn Mor hcal 
4 4 |} 3°45) 2-54) 19-06 | 1300. | 1216) 
3 | 57-70 8 | 5-99] 5-08] 21-19 | 1790 | 1586) 
16 - 111°07)10-16 | 25°47. | 2000 || F846) 

Rpg ty Soe a whe eee 
4 3°45} 2-54) 19-06 | 1370 sed 

4 | 80°23 8 4°8:99) 5:08}. 21210 1949 di 
11648) 


36 {11-07190°16) 25-47 2200 


And here it appears that the values of a, the first force of 
the charge are much more uniform and regular than by the 
former calculations in the preceding problem at least in all ex- 
cepting the smallest charge, 4 0z. in each gun ; which it would 
seem must be owing to some general cause or causes: Nor 
have we long to search, to find out what those causes may be. 
For when it is considered that these numbers for the value of 
#,in the last column of the table, ought to exhibit the first 
force of the fired powder, when it is supposed to occupy the 
gpace only in which the bare powder itself lies and that where- 
as it is manifest that the condensed fluid of the charge in these 
experiments, occupies the whole space between the ball and 
the bottom of the gun bore, or the whole space taken up by 
the powder and the bag or cartridge together, which exceeds. 
the former space, or that of the powder alone, at least in the 
proportion of the circle of the gua bore, to the same as dimi- 
nished by the thickness of the surrounding flannel of the bag 
that contained the powder; it is manifest that the force was 
diminished on that account. Now by gently compressing a 
number of folds of the flannel together, it has been found that 
the thickness of the single flannel was equal to the 40th part 
. of an inch; the double of which, 4, or -05 of an inch, is 
therefore the quantity ae which the diameter of the airele 

the 


400 PRACTICAL EXERCISES ON FORCES. 


- the powder within the bag, was less than that of the gun bore. 
But the diameter of the gun bores was 2-02 inches ; therefore, 
deducting the -05, the remainder 1:97 is the diameter of the 
powder cylinder within the bag: and because the areas of cir- 
eles are to each other as the spaces of their diameters, and 
the squares of these numbers, 1:97 and 2-02, being to each 
other as 388 to 408, or as 97 to 102; therefore, on this ac- 
count alone, the numbers before found, for the value of n, 
must be increased in the ratio ef 97 to 102. 


But there is yet another circumstance, which occasions the 
space at'first occupied by the inflamed powder to be larger 
than that at which it has been taken in the foregoing calcula~ 
tions, and that is the difference between the content of a sphere 
and cylinder. For the space supposed to be occupied at first 
by the elastic fluid, was considered as the length of a cylin- 
der measured to the hinder part of the curve surface of the 
- ball. which is manifestly too httle by the difference between 
the content of half the ball and acylinder of the same length 
and diameter, that is, by a cylinder whose length is 1 the se- 
midiameter of the ball. Now that diameter was 1-96 ania « ; 
the half of which is 0°98, and 1 of this is 0-33 nearly. Hence 
then it appears that the lengths of the cylinders at first filled 
by the dense fluid, viz. 3:45, and 5-99. and 11°07, have been 
all taken too little by 0°33 ; and hence it follows. that, on this 
account also, all the numbers before found for the wahie of . 
the first force n, must be further increased in the ratios of 3°45 
and 5:99 and 11-07, to the same numbers increased by 0:33, 
thatis, to the numbers 3°78 and 6°32 and 11-40. 


Compounding now these last ratios with the foregoing one, 
viz. 97 to 102, it produces these three, viz. the ratios of 334 
and 581 and 1074, respectively to 385 and 647. and 1163. 
Therefore, increasing the last column of numbers, for the 
value of n, viz. those of the 4 oz. charge in the ratio of 334 
to 385, and those of the 8 oz. charge in the ratio of 581 to 
647) and those; ot the: {6o7. (os gh a 
charge in the ratio of 1074 |Powder. The Guns. 





to 1163, with every gun, |—~ : SS 

they will be reduced to the 7 A oe an ces i ae ok 
~ numbers in the annexed ta- OZ. 

ble ; where the numbers 4 Aare: 1430 

are still larger and more re- 8  |1637|1677|1766/1812 


gular than before. 16 eae eae aa 


TNE Rig 


Thus 


PRACTICAL EXERCISES ON FORCES. 401 


Thus then at length it appears that the first force of the 
inflamed gunpowder, when occupying only the space at first 
filled with the powder, is about 1800, that is 1800 times the 
elasticity of the natural air, or pressure of the atmosphere 
in the charges with 8 oz. and 16 oz. of powder, in the two 
longer guns ; but somewhat less in the two shorter, probably 
owing to the gradual firing of gunpowder in some degree ; 
and also less in the lowest charge 4 oz. in all the guns, which 
may probably be owing to the less degree of heat in the small 
charge. But besides the foregoing circumstances that have 
been noticed, or used in the calculations, there are yet several 
others that might and ought to be taken into the account,.in or- 
der to a strict and perfect solution of:the problem ; such as, 
the counter pressure of the atmosphere, and the resistance of 
the air on the fore part of the ball while moving along the 
bore of the gun; the loss of the elastic fluid by the vent and 
windage of the gun; the gradual firing of the powder; the 
unequal density of the elastic fluid in the different parts of the 
space it occupies between the ball and the bottom of the bore ; 
the difference between pressure and percussion when the ball 
is not laid close to the powder ; and perhaps some others : on 
all which accounts it is probable that instead of 1800, the first 
force of the elastic fluid is not less than 2000 times the strength 
of natural air. 


Corol. From the theorem last used for the velocity of the 


‘ 2 
ball and elastic fluid, viz. v = 4/ Fs n <= = log. — ; cs 
af aoe ~log. ay we may find the velocity of the elas- 


tic Rata alone, viz. by taking w, or the weight of the ball, 
= 0 in the theorem, by which it becomes barely v = 


Oe Ip og. . for that velocity. And by computing 


the several seca examples by this theorem, supposing 
the value of n to be 2000, the conclusions come out a little 
various, being between 4000 and 5000, but most of them nearer 
to the latter number. So that it may be concluded that the 
velocity of the flame, or of the fired gunpowder expands it- 
self at the muzzle of the gun, at the rate of about 5000 feet 
per second nearly. 


Vor. U. 


Gh 
to 
© 
x 


402 MOTION OF BODIES IN FLUIDS - 


ON THE MOTION OF BODIES IN FLUIDS. 
PROBLEM XIX. 


Lo determine the Force of Fluids in Motion; and the Circum- 
stances attending Bodies Moving in Flucds. 


1. Ir is evident that the resistance to a plane, moving 
perpendicularly through an infinite fluid, at rest, is equal to 
the pressure or force of the fluid on the plane at rest, and 
the fluid moving with the same velocity, and in the contrary 
direction, to that of the plane in the former case. Buf the 
force of the fluid in motion, must be equal to the weight or 
pressure which generates that motion; and which, it is 
known, 1s equal to the weight or pressure of a column of 
the’ fluid, whose base is equal to the plane, and its altitude 
equal to the height through which a body must fall by the 
force of gravity, to acquire the velocity of the fluid > and 
that altitade is, for the sake of brevity, called the altitude 
due to the velocity. So that, if @ denote the area of the 
plane, 0 the velocity and n the specific gravity of the fluid ; 


2 
then the altitude due to the velocity v being ie the whole 
anv* 
4g" 
being 162, feet. And hence, ceteris paribus, the resistance 
is as the square of the velocity. 





i ; ‘ v2 
resistatice, or motive force m, will be aXnX-— = = 


. This ratio of the square of the velocity, may be other- 
wise derived thus. The force of the fluid in motion, must 
be as the. force of one particle multiplied by the number of 
‘them ; but the force of a particle is as its velocity ; and the 
number of them striking the plane in a given time, is also as 
the velocity ; therefore the whole force is as v X v or v?, that 
is the square of the-velocity. 


3. If the direction of motion, instead of being perpendi- 
cular to the plane, as above supposed, be inclined to it in 
any angle, the sine of that angle being s to the radius 1 
then se resistance to the plane, or the force of the fluid 
! against 


MOTION OF BODIES IN FLUIDS. 403 


against the plane, in the direction of the motion, as assigned 
above, will be diminished in the triplicate ratio of radius to 
the sine of the angle of inclination, or in the ratio of 1 to 53. 
For as being the direction of the plane, 
and sp that of the motion making the an- 
gle app, whose sine is s ;. the number of 
particles, or quantity ofthe fluid striking 
the plane, will be diminished in the ratio Re 
of 1 to s, er of radius to the sine of the 
angie s of inclination; and the force of 
each particle will also be diminished in the same ratio of 1 to 
s: so that on both these accounts, the whole resistance will 
be diminished in the ratio of 1 to s?, or in the duplicate ratio 
ef radius to the sine of the said angle. But again, it is to be 
considered that this whole resistance is exerted in the direc- 
tion BE perpendicular to the plane; and any force in the di- 
rection Bg, is to its effect in the direction ar, parallel to Bp, 
as AE to BE, that is as 1 to s.- So that finally, on all these:ac- 
counts, the resistance in the direction of motion, is diminished 
in the ratio of 1 to s?, or in the triplicate ratio.uf radius to the 
sine of inclination. Hence comparing this with article 1, the 
whole resistance, or the motive force on the plane, will be m 
anv? $3 





4. Also, if w denote the weight of the body, whose plane 
face @ 1s desir by the absolute force m; then the retarding 
anv2s 


force f, or— will be — 
Acw © 





5. And.if the body be a cylinder, whose face or end is a? 
and diameter d, or radius, r, moving in the direction of its 
axis; because then s = 1, and a = pr? —1pd?, eben P = 
3°1416 : the resisting force m will be - 

NHnd2n2 2 2 - : 
Sa es dnl and the retarding force f = _npdte’ _ap a 
l6g 4g l6gw 4ow 





6. This is the value of the resistance when the end of the 
cylinder is a plane perpendicular to its axis, or to the direc- 
tion of motion. But were its face a conical surface, or an 
elliptic section, or any other figure every where equally in- 
clined to the axis, the sine of inclination being s: then the 
- number of particles of the fluid striking the face being still the 


same but the force of each, opposed to the direction of mo- 
fion, 


404 MOTION OF BODIES IN FLUIDS. 


tion, diminished in the duplicate ratio of radius to the sine of 





inclination, the resisting force m would be - Sh - 
mpd? ns? _ npr? 0? s? 
“16g m4 4g 


But if the body were terminated by an eud or face of any 
other form as a spherical one, or such like, where every part 
of it has a different inclinatidn to the axis; then a further 
investigation becomes necessary, such as in the ier bet pro- 
position. 


PROBLEM XX. 


To determine the Resistance of a Fluid to any Body, moving in 
it, of a Curved End; as a Sphere, or a Cylinder with a He- 
mispherical End, &c. 


1. Let BEAD be a section through the 
axis ca of the solid, moving in the direc- 
tion of that axis. To any point of the 
curve draw the tangent ec, meeting the 
axis produced in Gc: also, draw the per- 
pendicular ordinates er, indefinitely 
near each other ; and draw ae parallel 
to cG. 





Putting cr = 2, EF = y, BE== 2, s=sine Zc to radius 
1, and p, = 3-1416: then 2py is the circumference whose 
radius is Er, or the circmference deseribed by the point ©, 
in revolving ‘about the axis ca ; ; and 2py x xe or 2pyz is the 
fluxion of the surface, or it is the surface described by ze, 
in the said revolution about ca, and which is the quantity 
represented by a in art. 3 of the last problem: hence 
a0 8 prev 

“4g X Qpyz or 
or the fluxion of the resistance to the body, whatever the 
figure of it may be. And the fluent of which will be the re- 
sistance required. | 








p23 
9 rte yz is the resistance on that ring, 


2. In the case of a spherical form: putting the radius ca 


—_——— EF CF x 
or cB = 7, we have y= 4/r? — 27, s = — = — =~ and 
EG CE r 


yz, OF EF XEC== cE X ae =rx; therefore the general 


. pnv? Weve. fe nv2 3 ; 2 
fluxion 2 x s?yz becomes #—— x ~—X re = PE 
e. 9 2g r3 Qera 














7 a 


the 


MOTION OF BODIES IN FLUIDS. 405 


2 
the: fluent of which, or a4, is the resistance to the spe- 


rical surface generated by se. And when & or c¥F is =r or ca, 


i novi 72 
it becomes Se 
8c 





for the resistance on the whole hemis- 

Pex pnv? d? | : 
phere ; which is also equal to “Bag” where d=2r the dia- 
meter. | 


3. But the perpendicular resistance to the circle of the 
same diameter d or sp, by art. 5 of the preceding problem, 
ag Pav a? 
l6g ; 
resistance to the sphere, is just equal to half the direct resist- 


ance to a great circle of it, or to a cylinder of the same dia- 
meter. 





; which, being double the former, shows that the 


4. Since 1pd? is the magnitude of the globe; if nw denote 

its density or specific gravity, its Metent w will be = Ipd?y, 
2 2 6 

and therefore the rotate force f or— ity Sear Mira = 
nv? 325 pnd 

eend? which is also = ee by art. 8 of the general theorems 





; 3n i ite 
: ites al pe aa 8g P 
in page 380 ; hence then ae and s cs X 4d; which is 


the space that would be described by the globe, while its whole 
motion is generated or destroyed by a constant force which is 
equal to the force of resistance, if no other force acted on the 
globe to continue its motion. And if the density of the fluid 
were equal to that of the globe, the resisting force is such, 
as, acting constantly on the globe without any other force, would 
generate or destroy its motion in describing the space ?d, or 4 
of its diameter, by that accelerating or retarding force. 


5. Hence the greater velocity that a globe will acquire by 
descending in a fluid, by means of its relative weight in the 
fluid, will be found by making the resisting force equal to that 
weight. For, after the velocity is arrived at such a degree, 
that the resisting force is equal to the weight that urges it, it 
will increase no longer, and the globe will afterwards conti- 
nue to descend with that velocity uniformly. Now, n andn 
being the separate specific gravities of the globe and fluid, 
n — 7 will be the relative gravity of the globe in the fluid, 
and therefore w = 1pd? (n — n) is she weight by which ay is 

urge 


406 MOTION OF BODIES IN FLUIDS, 


| pu? ds <. a 
urged; also m saeaa? Pass the resistance; eonsequently 





2 2 
p so ==1pd3 (w — n) when the velocity becomes uniform : 
from which equation is found v=,/ (4¢ . 4d. ee for the. 


said uniform-or greatest velocity. 


And, by comparing this form with that in art. 6 of the gene- 
ral theorems in page 379, it will appear that its greatest velo- 
city, 1s equal to the velocity generated by the accelerating 





N—w . are 
force , in describing the space 4d, or equal to the velo- 


city generated by gravity in freely describing the space a 
Add. If n=2n, or the specific gravity of the globe be dou- 
ble that of the fluid, then Ee] == the natural force of gra- 


vity ; and then the globe will attain its greatest velocity in de- 
scribing $d or 4 of its diameter.—It is further evident, that 
if the body be rery small, it will very soon acquire its great- 
est velocity, whatever its density may be. . 


Exam. If a leaden ball, of 1 inch diameter, descend in wa- 
ter, and in air of the same density as at the earth’s surtaess 
the three specific oe bab as 11%, and 1, and 25% 5: 
Than v = ,/4. 16), .;4,- 101==1,/31 . 193 = 8'5944 feet, 
is the greatest velocity per. ascot the ball can acquire by de- 


scending in water. And v=,/4. 488 . 4. 34 259° nearly 
S59 4/ 347493 == 259-82 is the greatest alockiy it.can acquire 
In air. 


But if the ‘eich were only jj}, of an inch diameter, the 
greatest shemale i it could acquire, would be only 7 q's Of these, 
namely ,£5 of a foot in water, and 26 feet nearly in air. And 
if the ball were still further diminished, the greatest velocity 

would also be diminished, and that in the subduplicate ratio 
of the diameter of the ball. 3 


PROBLEM XXI. 


To determine the Relations of Pe elocity, Space, and Time, of a 
Ball moving in a aioe in'which it is projected with a Given 


Hehaeity 
1. Ler 


MOTION OF BODIES IN FLUIDS. 407 


1. Ler a=the first. velocity of projection, « the space 
described in any time ¢, and v the velocity then. Now, by 


5 3nv? 
art .4 of the last problem, the accelerative force f = ca > 


where n is the density of the fluid, n, that of the ball, and 
d its diameter. . Therefore the general equation vy = 2g¢f 5 


becomes Be a a ad at Te a em Fie 
—3nv2 : —3n. 3n 
ey 2 3 and hence — agree oo — by putting b mn roe 


The correct fluent of this, is log. a — log. v OF to -— bx. 
Or, putting c <= 2-718281828, the number whose hyp. log. 


a 
is 1, then1 is < ~—= c*, and the velocity a Diao a aM are 


. The velocity v at any time haley the c—* part of the 
ate velocity, therefore the ee lost in any time, will be 


cbx— 
the 1 —c—»* part, or the 





sot of the first velocity. 


EXAMPLES, 


Exam. 1. If a globe be peared ida. with any velocity, in a 
medium of the same density with itself, and it deseribe a 
space equal to 3d or 3 of its diameters. Then « = 3d, and 


3n 3 aot et 2:08 . 
bese 7 = Gp therefore br e= 3, and —— ae Sie is the ye- 


locity lost, or nearly 2 of the projectile velocity. 


Exam. 2. If an iron ball of 2 inches diameter were pro- 
jected with a velocity of 1200 feet per second ; to find the © 
velocity lost after moving aronEh any space, a8 suppose 500 
feet of air: we should have d= 2, =}, a = 1200, « == 500, 





N= 71, = 0012; and therefore br = - a il - 
ong. 3.12 .500.3.6 81 00 i 
xd 8.22.10000. 440° a Stead 


second: having lost 202 feet, or nearly 3 of its first velocity. 


_ Exam. 3. If the earth revolved about the sun, in a me- 

dium as depse as the atmosphere near the earth’s surface ; 

and were required to find the quantity of motion lost in a 
year, 


408 MOTION OF BODIES IN FLUIDS. 


year. Then since the earth’s mean density is about 43, and 
its distance from the sun 12000 of its diameters, we have 
24000 X 3.1416 = 75398 diameters = x, and ares res Meg 


3-75398 12.2 oi 
= 7:5398 ; h 
“$. 10000. 9 2B 7 RENCE 


of the first motion in the space of a year, and only the ,3';4 
part remains. 





== 1880 parts are lost 


Exam. 4. If it be required to determine the distance 
moved, xz, when the glebe has lost any part of its motion, as 
suppose 4 and the density of the globe and fluid equal ; 

ies : , I Bd. sss 
The general equation gives « = ; X log. <= = X log. of 

1°8463925d. So that the globe loses half its motion be- 
fore it has described twice its diameter. 








. , : : bs 
3. To find the time t; we have z =- == —" 
Now to find the fluent of this, put z = c*; then is br = 
log. z, and bz = z or xz a conseq. ¢ or OP Trees 
g 2; AT ea vaste PM q:. ¢ aC akg 
; bx 
ee ee — and hence t == A= But as ¢ and 2x vanish 
a ab ab ab 
| ION peat 1 
together, and when zc = 0, the quanity rT 
therefore, by correction ise : pt os t 
oe aasae LTB bh ba eee 
; on a . 
the time sought ; where 6 = ade and v = —,the velocity 


Exam. If an iron ball of 2 inches diameter were projected 
in the air with a velocity of 1200 feet per second ; and it 
were required to determine in what time it would pass over 
50U yards or 1500 feet, and what would be its velocity at the 
end of that ttme: We should have, as in exam. 2 above, 


3.12.3 65M ‘6 21500 875 
am) = fe cme ce eh meaty, Seat 
8.12.10000 2716 * 2716 679”. 
12716 NL oleh ak eg 
beak es oe. Feat Seok. 1200 «690 


Me Consequently v = 690 is ‘ate velocity; and ¢ = 


© 1, S| 
AN oes 
ee gett te x aan 7200 


“a 
atria: or 1" and 3 nearly. 


~\ = > 12) 1 seconds, in the time 


PROBLEM 


. MOTION OF BODIES IN FLUIDS, 409 


- PROBLEM XXII. 


To determine the Relitiohe of Space, Time, and Velocity, when 
a Globe descends, by its own Weight, in a Fluid. 


Tue foregoing notation remaining, viz. d = diameter, 
nandn the density of the ball and fluid, and v, s, ¢, the 
velocity, space, and time, in motion ; we have 1nd3 = the 
magnitude of the ball, and 1pd? (n—n) = = its weight i in the 


d2y2 
=? — ye — = its resistance from the fluid ; con- 


pnd? 
ae = is the motive force by 


which the ball is urged ; 3, being divided by 1 wd?, the 


2 
~—— a sohald for the 
N 16gnd 


fluid, also m= 








_ sequently tpd5 (n—n) _ 


quantity of matter moved, gives f= 1— 


accelerative force. 
2. Hence vo = 2¢ fs, and s = radar ees et ie 
2of 3n . 
2g (N — fi)? 


Pe co ta 3n 
5 Oca sa? putting 6 =~ aap and <= or ag Bd ay 


8 nearly ; the ae of which iss = - Eat 
7% log. of - 


ie inci ¢ Vs Tephat is, when s and v begin, or are equal to 
nothing, both together. 


, or ab== 





an expression for the space s in terms of 


But if the body commence motion in the fluid with a cer- 
tain given velocity e, or enter the fluid with that velocity, like 
as when the body, after falling in empty space from a certain 
height, falls into a fluid like water ; then the goprect fluent will 


be s=,, X hyp. log. ee 


3. But now, to determine v in terms of s, put ¢c = 


2°718281828 ; then since the log. of fz 3 == 2bs, therefore 





a a—v? 
=o, or = pean ; hencé v= = <%= - <8 
a 





a—v? 
a—ac~—s, ig the velocity sought. | 
VT, ie is eee 4. The 


us 


20, by sey fs i-"— 


(2g . 8 


410» MOTION OF BODIES IN FLUIDS, , 


4. The greatest velocity is to be found, as in art. 5 of prob. 
3nv2 


iOgnd = 0, which gives v= 











The same value of v is obtained by 


making the Raabe of v2, or of G—ac—%s, = Q, ‘And the same 


value of v is also obbaned by making s infinite, for then c—?>s 


==0. But this velocity 4/a cannot be attained in any finite time. 
and it only denotes the velocity to which the general value of 
v Or 4/ a ac 2bs continually approaches. It is evident how- 
ever, that it will approximate towards it the faster, the greater 
bis, or the less dis; and that the diameters being very small, 
the bodies descend by nearly uniform velocities, which are 
direct in the subduplicate ratio of the diameters. See also 
art. 5, prob. 20, for other observations on this head. 


5. To find the time t. Now 7 =~ = Se beans een 
: a Jia 
Then, to find the fluent of this ae put z=/1—c-s 





Zz 


~ 


eth 
==, or z22==1 —c—**§- hence zz=d¢ c—*>*, and ¢== 
a 














be—2bs 
1. £2 consequently #=— 2, 
Bee eee De ud ees | 
. ; : 1 Ee ts: 1 
and therefore the fluent igs x log pis ya 
es we abs ral 
4 log 1+ 1 ae X lo Jfatu 


eens . “——., which is’ th 
Vf 188 2b s/o 5 fad 4 re a se 
general expression for the time. i 

Exam. If it were required to determine the time and velo- 


city, by descending in air 1000 feet, the ball ve of dead, 
and 1 inch diameter. : 


Here n= lili, a=s255 je ara anil s=1000.. 
Hence a=———. loads: Mp2 198.8. 84 7 34 . 2500 = 
| 3 ahre $2812 018.3 
193 . 34. 502. 35 rie (363012. 9..9 
ie. 87 ee Ti}. =F 8. 34. 2500 68.502° 
consequently v= Vax /l— qo = 188-84 808 X 


Soup - > 


(1 a chs )= 2032 the velocity. And ¢= —— 5 


bfa 


? 


MOTION OF BODIES IN FLUIDS. 4\} 





4. /{ et  ) § 34.9500 2 . 478383 

+/1- c pi x log. cdi hada 4 8-5236", 
imyin 27.193 0-21617 
the time. } 


Note. If the globe be so o light as to ascend in the fluid ; it 
is only necessary to change the signs of the first two terms 
in the value of f, or the accelerating force, by. which ‘it be- 


n 3102 
comes f =——1-—--———-;; and then roceedin in all res ects 
i} N l6gnd’ P S ged 
as before, | 
SCHOLIUM. 


To compare this theory, contained in the last four problems, 
with experiment, the few following numbers are here extract- 
ed from extensive tables of velocities and resistances, result- 
ing from a course of many hundred very accurate experiments, 
made in the course of the year 1786. 


In. the first column are contained the mean uniform or great- 
est velocities acquired in air, by globes, hemispheres, cylin- 
ders, and cones, all of the same diameter, and the altitude of 
the cone nearly equal to the diameter also when urged by the 
several weights expressed in avoirdupois ounces, and stand- 
ing on the same line with the velocities, each in their proper 
column. So, in the first line, the numbers show, that when 
the’greatest or uniform velocity was accurately 3 feet per se- 
cond, the bodies were urged by these weights, according as 
their different ends went foremost ; namely, by :028 oz. when 
the vertex of the cone went foremost ; by :064 oz. when the 
base of the cone went foremost; by -027 oz. for a whole 
sphere ; by 050 oz. for a cylinder ; by 051 oz. for the flat 
side of the hemisphere ; and by :020 oz. for the round or 
convex side of the hemisphere. Also at the bottom of. all, 
are placed the mean proportions of the resistances of these 
figures in the nearest whole numbers. Note, the common di- 
ameter of all the figures, was 6-375, or 62 inches ; so that the 
area of the circle of that diameter is just 32 square inches or 
2.of a square foot; and the altitude of the cone was 63 inches. 
Also, the diameter of the small hemisphere was 42 inches, 
ane consequently the area of its base 172 square inches, or. 

Lofa square foot nearly. 

: * From the given dimensions of the cone, it appears, that the 
angle made by its side and axis, or direction of the path, is 26 


degrees, very nearly. 
. eh The 


412 MOTION OF BODIES IN FLUIDS: 


The mean height of the barometer at the. times of making 
the experiments, was nearly 30-1 inches, and of the thermo- 
meter 62° ; consequently the weight of a cubic foot of air 
‘was equal to 11 oz. nearly in those circumstances. 








vertex.|base. 


Veloc. One: \Whole Cylin- Hemisphere. ele 
gobe.| der. emis. 


aaaed flat. jround | flat. 


























feet. oz. | OZ. OZ. OZ. OZ. oz. OZ. 


028 | -064] -027 | :050 051| 020] :028 

4 | -048°| +109) -047 | -090 096} -039) -048 
5 | O71 | 162] -068 | +143 148} 063) +072 
6 | 098 | +225) -094 | +205 211} +092) +103 
7 | +129 | -298} -125 | -278 | +984] -123| +141 
8. | -168 | -382| -162 | -360 368] +160} -184 
9 | -211 | -478] 205 | -456 “464; +199] +233 
10 | *260 | °587| -255 | -565 "B73. 2942 OBF 
11. } -315 | -712) -310 | -688 -698} -297| +349 
12 | +376 | -850) -370 | +826 836]. °347; +418 
~13 | +440 1-000} -435 | -979 -988} -409]  -492 
14 | +512 |1-166} -505 {1-145 | 1-154) +478) = -573 
15 | °589 |1°346} +581 |1°327 | 1-336] +552! +661 
16 | *673 |1°546} -663 [1-526 | 1-538) +634) -'754 
17. | *762 -|1-763) +752 (1-745 | 1-757) +722] 853 
18 | ‘858 [2-002] -848 |1-986 }|- 1-998] +818]. -959 
19 | -959 |2-260) -949 |2-246 | 2-258] +922) 1-073 
20 |1:069 |2-540/1-057 |2-528 | 2-542) 1033} 1:°196 


eomemmeres | cecum | eee te | eeereeerneees | eee oeressecs | eer 


1 496! 291] 124 | 285 | 988 119 | 140 
Numb. 








From this table of resistances, several practical inferences . 

may be drawn. As, : | 
1. That the resistance is nearly as the surface ; the resist- 
ance increasing but a very little above that proportion in 
the greater surfaces. Thus, by comparing together the num- 
bers in the 6th and last columns, for the bases, of the two 
hemispheres, the areas of which are in the proportion of 
172 to 32, or as 5to9 very nearly ; it appears that the 
numbers in those two columns, expressing the resistances, 
are nearly as i to 2, or as 5 to 10, as far as to the velocity 
of 12 feet; after which the resistances on the greater sur- 
face increase gradually more and more above that propor- 
tion. And the mean resistances are as 140 to 288, or as 5 
: to 


MOTION OF BODIES IN FLUIDS. 413 


io 102. This circumstance therefore agrees nearly with the 
theory. 


2. The resistance to the same surface, is nearly as the 
square of the velocity ; but gradually increasing more and 
more above that. proportion, as the velocity increases. This 
is manifest from all the columns. -And therefore this circum- 
stance also differs but little from the theory, in small veloci- 
ties. 


3. When the hinder parts of bodies are of different forms, 
the resistances are different, though the fore parts be alike ; 
owing to the different pressures of the air on the hinder parts. 
Thus, the resistance to the fore part of the cylinder, is less 
than that on the flat base of the hemisphere, or of the cone ; 
because the hinder part of the cylinder is more pressed or 
pushed, by the following air, than those of the other two fi- 
gures. 


4. The resistance on the base of the hemisphere, is to that 
‘on the convex side nearly as 22 to 1, instead of 2 to 1, as the 
theory assigns the proportion. And the experimented resist- 
ance, in each of these, is nearly } part more than that which 
is assigned by the theory. 


5. The resistance on the base of the cone is to that on the 
vertex, nearly as 2;% to 1, And in the same ratio is radius 
to the sine of the angle of the inclination of the side of the 
cone, to its path or axis. So that, in this instance, the resist- 
ance is directly as the sine of the angle of incidence, the 
transverse section being the same, instead of the square of the 
sine. 


6. Hence we can find the altitude of a column of air whose 
pressure shall be equal to the resistance of a body, moving 
through it with any velocity. Thus, 

Let a = the area of the section of the body, similar to any 
of those in the table, perpendicular to the direction 
of motion ; 

vy =the resistance to the velocity, in the table; and 
x = the altitude sought, of-a column of air, whose base 
‘is a, and its pressure r. 

Then ax = the content of the column in feet, 

and 1laa or faz its weight in ounces; =- - - - - = = 


therefore Sar =r, and ¢ = 8 x “is the altitude sought in 
a 


feet, 


414 MOTION OF BODIES IN FLUIDS. 


feet, namely, & of the quotient of the resistance of any body 
divided by its transverse section: which is a constant quan- 
tity for all similar bodies, ‘however different in magnitude, 
since the veerstance ris as the section a, as was found in art. 1. 

“When a== 2 ofa foot, as in all the figures 1 in the ss 


ing table, except the small hemisphere : then, ee x - — 


becomes x = 15r, where r is the resistance in the table, i 
the similar body. 


If, for example, we take the convex side ‘of the large he- 
misphere, whose resistance is ‘634 oz. to a velocity of 16 feet 
per second, then r = 684, and « = 1 r = 2°3775 feet, is the 
altitude of the column of air whose pressure is equal to the 
resistance on a spherical surface, with a velocity of 16 feet. 
And to compare the above altitude with that whichis due to the 
given velocity, it will be 32? : 162 :: 16: 4, the altitude 
due to the velocity 16 ; which is near double the altitude that 
is equal to the pressure. And as the altitude is proportional 
to the square of, the velocity, therefore, in small velocities, 
the resistance to any spherical surface is equal to the Pi genay 
of a column of air on its great circle, whose altitude is 12 or 

594 of the altitude due to its velocity. 


But if the cylinder be taken, whose resistance r = 1:526: 
theng = 1 r = 5:72; which exceeds the height, 4, due to 
the velocity i in the, ratio of 23 to 16 nearly. And the differ- 
ence would be still greater, if the body were fryer j and also 
if the velocity were more. . 


7. Also, if it be required to find with what velocity any flat 
surface must be moved, so as to suffer a resistance just equal 
to the whole pressure of the atmosphere : 

The resistance on the whole circle whose area is 2 of a foat, 
is °051 oz. with the velocity of 3 feet per second ; itis i of - 

‘051, or ‘0056 oz. only with a velocity of 1 foot. But Qh x 
13600 K 2 = 75553 oz. is the whole restive of the atmos- 
phere. Therefore, as ,/0056: \/ 7556 :: 1: 1162 nearly, 
which is the velocity sought.. Being almost nee to the velo- 
city with which air rushes into a Yacuyp, 


8. Hence may be jiferted the great resistance suffered by 
military projectiles. For in the table, it appears, that a globe 
of 63 inches diameter which is equal to the size of an iron ball 
weighing 36lb, moving with a velocity of only 16 feet per se- 
cond, meets with a resistance equal to the pressure of 2 of an 
ounce weight; and therefore, POMEROY only according to the 

square 


% 


MOTION OF BODIES IN FLUIDS. 416 


square of the velocity, the least resistance that such a ball 
would meet with, when moving with a velocity of 1660. feet 
would be equal to the pressure of 4171b, and that independent 
of the pressure of the atmosphere itself on the fore part of the 
ball which would be 487lb more, as there would be no pres- 
sure from the atmosphere on the hinder part, in the case of so 
great a velocity as 1600 feet per second. So that the whole 
resistance would be more ihan 900lb to such a velocity. 


9. Having said, in the last article, that the pressure of the 
atmosphere is taken entirely off the hinder part of the ball 
moving with a velocity of 1600 feet per second; which must 
happen when the ball moves faster than the particles of air 
can follow by rushing into the place quitted and left void by 
the ball, or when the ball. moves faster than the air rushes 
into a vacuum from the pressure of the incumbent air; let 
us therefore inquire what this velocity is. Now the velocity 
with which any fluid issues, depends on its altitude above 
the orifice, and is indeed equal to the velocity acquired by a 
heavy body in falling freely through that altitude. But, sup- 
posing the height of the barometer to be 30 inches, or 21 feet, 
the height of a uniform atmosphere, all of the same density as 
at the earth’s surface, would be 21} % 13:6 x 8331 or 28333 
feet; therefore,/16 : ,/28333 : : 32 : 8,/28333=1346 feet, 
which is the velocity sought. And therefore, with a velocity 
of 1600 feet per second, or any velocity above 1346 feet, the 
ball must continually leave a vacuum behind it, and so must 
sustain the whole pressure of the atmosphere on its fore part, 
as well as the resistance arising from the vis inertia of the 
particles of air struck by the ball. 


10. On the whole, we find that the resistance of the air, as 
determined by the experiments, differs very widely, both in 
respect to its quantity on all figures, and in respect to the pro- 
portions of it on oblique surfaces, from the same as determin- 
ed by the preceding theory: which is the same as that of Sir 
Isaac Newton, and most modern philosophers. Neither should 
we succeed better if we have recourse to the theory given by 
Professor Gravesande, or others, as similar differences and 
inconsistencies still occur. | 


We conclude therefore, that all the theories of the resist- 
ance of the air hitherto given, are very erroneous. And the 
preceding one is only laid down, till further experiments, on 
this important subject, shall enable us to deduce from them 
another, that shall be more consonant to the true phenomena 
of nature. a Oi eer 


Mckee ON 


[ 416 ] 


ON THE MOTION OF MACHINES, AND THEIR 
MAXIMUM EFFECTS. 


Arr. 1. When forces acting in contrary directions, or 
in any such directions as produce contrary effects, are ap- 
plied to machines, there is, with respect to every simple ma- 
chine (and of consequence with respect to every combination 
of simple machines) a certain relation between the powers 
and the distances at which they act, which, if subsisting in 
any such machine when at rest, will always keep it in a state 
of rest, or of statecal equilibrium; and for this reason, be- 
cause the efforts of these powers when thus related, with 
regard to magnitude and distance, being equal and opposite, 
annihilate each other, and have no tendency to change the 
state of the system to which they are applied. So also, if 
the same machine have been put into a state of uniform mo- 
tion, whether rectilinear or rotatory, by the action of any 
power distinct from those we are now considering, and these 
two powers be made to act upon the machine in such motion 
in a similar manner to that in which they acted upon it when 
at rest, their simultaneous action will preserve it in that state 
of uniform motion, or of dynamical equilibrium : and this for 
the same reason as before, because their contrary effects de- 
stroy each other, and have therefore no tendency to change 
the state of the machine. But, if at the time a machine is 
in a state of balanced'rest, any one of the opposite forces be 
increased while it continues to act at the same distance, this 
excess of force will disturb the statical equilibrium, and pro- 
duce motion in the machine ; and if the same excess of force 
continues to act in the same manner, it will, like every con- 
stant force, produce an. accelerated motion; or if it should 
undergo particular modifications when the machine is in dif- 
ferent positions, it may occasion such variations in the motion 
as will render it alternately accelerated and retarded. Or the 
different species of resistance to which a moving machine is 
subjected, as the rigidity of ropes, friction, resistance of the 
air, &c. may so modify a motion, as to change a regular or ir- 
regular variable motion into one which is uniform. 

2. Hence then the motion of machines may be considered 
as of three kinds.. 1. That which is gradually accelerated, 
which obtains commonly in the first instants of the commu- | 
nication. 2. That whichis entirely uniform. 3. That which 
is alternately accelerated and retarded. Pendulum clocks, 


and machines which are moved by a balance, are related to 
the 


MAXIMUM IN MACHINES. 417 


the third class. Most other machines, ashort time after their 
motion is commenced, fall under the second. Now though 
the motion of a machine is alternately accelerated and retard- 
ed, it may, notwithstanding, be measured by a uniform motion, 
because of the periodical and regular repetition which may 
exist in the acceleration and retardation. Thus the motion of 
a second’s pendulum, considered in respect to single oscilla- 
tion, is accelerated during the first half second, and retarded 
during the next: but the same motion taken for many oscilla- 
tions may be considered as uniform. Suppose, for example, 
that the extent of each oscillation is 5 inches, and that the pen- 
dulum has made 10 oscillations : its total effect will be to have 
Fun over 50 inches in 10 seconds; and, as the space describ- 
ed in each second is the same, we may compare the effect to 
that produced by a moveable which moves for 10 seconds with 
a velocity of 5 inches per second. We see, therefore, that 
the theory of machines whose motions are uniform, conduces 
naturally to the estimation of the effects produced by machines 
whose motion is alternately accelerated and retarded : so that 
the problems comprised in this chapter will be directed to 
those machines whose motions fall under the first two heads ; 
such problems being of far the greatest utility in practice. 


Defs. 1. When ina machine there is a system of forces or 
of powers inutually i in opposition, those which produce or tend 
to produce acertain effect are called, movers or moving pow- 
ers; and those which produce or tend to produce .an ” effect 
which opposes those of the moving powers, are called reszst- 
ances. If various movers act at the same time, their equiva- 
lent (found by means of prob. 7, Motion and Forces) i is called 
individually the moving force ; and, in like manner, the result- 
ant of all the resistances reduced to some one point, the re- 
sistance. This reduction in all cases simplifies the investiga- 
tion. | 


2.. The impelled point of a machine is that to which the ac- 
tion of the moving power may be considered as immediately 
applied ; and the working point is that. where the resistance 
arising from the work to be performed immediately acts, or to 
which it ought all to be reduced. Thus, in the wheel and 
axle, (Mechan. prop. 32), where the moving power P is to 
overcome the weight or resistance w, by the application of 
the cords to the wheel and to the axle, 8 is the impelled point, 
and a the working point. : 


Vou: tks Cae 3, The . 


A418 MAXIMUM IN MACHINES. 


. The velocity of the moving power is the same as the ve- 
tacit of the impelled point; the velocity of the resistance ‘the 
same as that of the working point. 

4. The performance or effect of a machine, or the work dois 
is measured by the product of the resistance into the velocity — 
of the working point ; the momentum of impulse is measured — 

by the product of the moving force into the velocity of the 
impelled point. 

These definitions being established we may now exhibit a 
few of the most useful problems, giving as much variety In 
their solutions as may render one or other of the methods of 
easy application to any other cases which may occur. 


PROPOSITION I. 


if n, and r be the distances of the power P, and the weight ¢ or 
resistance w, from the fulcrum ¥ of a straight lever: then wili 
the. relat of the power and of the weight at the end of any 
2p—Rrw pihtre es r?w 
Re R2p-+r pares f R2 pre w 
anil inertia of the lever itself not being te sapahas 
If the effort of: the power ba- Va x 
lanced that of the resistance, P 


y ‘ > 
would be equal to—. Consequent- be Wi fi 


ly, the difference between this value of p, and its actual value, 


time thee * gt, respectively, the weight 


or P — ~ Ww, will be the force which tends to move the lever. 


And because this power applied to the point a accelerates the 
masses P and w, the mass to be substituted for w, in the point 


2 ‘ 

a, must be = W, (Mechan. prop. 50) in order that this mass 
at the distance R may be equally accelerated with the mass 
: 9 
w at the distance rk. Hence the power p — a will accele 
rate the quantity of matter Pp +- “ w; and the accelerating 


, re PR? —Rrw 
force F ==(r ——w)+(e-+- EW errcere 7 But (Art 33,Gen. 
Laws of Motion) v o Ft or is = gir (g being = 323 feet) ; 
R?2P—Rrw 

n2p--r2w 
because veloc. of vp: veloc. of w::k: tr, pherefare veloc. of 
_t ae RR ll P—Rrw RTP — 72 Ww ; 
% rita : KR ReP+erawe “Rap-pewe. 











which: in this case = gt, the velocity of ep. And, 








Corol. 


MAXIMUM IN MACHINES. 419 


Corol. 1, The space described by the power in the time ¢, 
R2P=—REW 


will be = rca ly 


zg? ; the space described by w in the 


RrP—?r2Ww 


same time will be = Lof?, 
R?p-R? Ww 38 





Cor. 2. lfrir:: 1, then will the force lebaite acce- 
Pn? —wn 


lerates a be = : 
“Pn? pn? + w Ww 

Cor.3. If at the same time the inertia of the moving 
force P be = 0, as in muscular action, the force accelerating 


Pn? — Wi 
willbe =<: 
WwW 


Cor. 4. If the mass moved have no weight, but possesses 
inertia only, as when a body is moved along a aipriconiel 
plane, the force which accelerates a will be = a ce . And— 


either of these values may be readily intiodricad into the in- 


vestigation. 


Cor. 5. The work done in the time ¢, if we retain the ori- 
: ae 2 RrP—r?w -  RrPwW—Pr2 w2 
ginal notation, will be = ——-—— pa RENN Te ARG 
eR R2p-+r?w 


Cor. 6. When the work done is to be a maximum, and we 
wish to know the weight when Pp is given, we must make 
the fluxion of the last expression =0. Then we sen pave, 


4 
eR3p2 — 972 p2 pw yt w2=Oandw=pP X Men =a): 


Cor.7. If nk :r::n: 1, the preceding expression will 
become w = Pp X [,/(n4 +n?) —n? ]. | 


Cor. 8. When the arms of the lever are equal in length, 
that is, when n = 1, then is w= P X.(,/2- eae = :414214p, 
or nearly 4%, 4, of the moving force. 

Scholium. 


If we in like manner investigate the formule relating to 


' motion on the axis in peritrochio, it will be seen that the 


expressions correspond exactly. Hence it follows, that when 
itis required to proportion the power and weight so as to 
obtain 


420 MAXIMUM IN MACHINES. 


obtain a maximum effect on the wheel and axle, (the weight 

of the machinery not being considered), we may adopt the 
conclusions of cors. 6 and 7 of this prop. And in the extreme 
case where the wheel and axle becomes a pulley, the expres- 
sion in cor. 8 may be adopted. The like conclusions may be 
applied to machines in general, if rn and 7 represent the dis- 
tances of the impelled and working points from the axis of 
motion ; and if the various kinds of resistance arising from 
friction, stiffness of ropes, &c. be properly reduced to their 
equivalents at the working points, so as to be comprehended 
in the character w for resistance overcome. 


PROPOSITION II. 


Given R and r, the arms of a straight lever, m and m their re- 
spective weights, and Pp, the power acting at the extremity of the 
arm Rk ; to find the weight raised at the extremity of the other 
arm when the effect is a maximum. 

In this case 1m is the weight of : 

2 £ A ; FEF B 


the shorter end reduced to B, and _ : 
conseq. —— is the weight which cP : wf 
applied at a, would balance the shorter end; therefore 


sete —w, would:sustain both the shorter end and the weight 
Or 


w in equilibrio. Put e + im is the power really ssid at i 
longer end of the lever ; consequently. - - 


P oe ymin (or —w); i is the absolute moving power. Now 


the aintshee ‘ef the centre of gyration of the beam from r* 





* The distance of R, the centre of gyration, from © the centre or axis of motion, 
in some of the most useful cases, is as below ; 


; : é Wr t 
in a circular wheel of uniform thickness - - - - 70° + - R= rail. /2 
1 
Tn the periphery of a circle revolving about the diam. - - - cr=rad. fz 
In the plane of acircle’ - - - '- + - ditto -°+ - - + cr=J rad. 
In the surface of a sphere -_- - - - ditto - - - - -. eR ~-jad, of 2 3 
Ina solid sphere -- < - -‘- + - (= ditto - - -‘+ + cr==rad. v3 
Ina plane ring formed of circles whose radii are Ry 2 cr yy — RA 
revolving about centre (=<) (o.oo ye ne fe 2nr3—2r2 
In a cone revolving about its vertex. - = - - = - + crely 12 024272, 
Inacone’- - - - - itsaxis -°- -- + - + - CR= Pf 3B 
I bet Repo vols ries na $73 
na straight lever whose arms are R as Pe es Vie alii = MH Fara Cir)”, 


is 


MAXIMUM IN MACHINES. 421 


Re+ 2, 
“> ¥ SG Bee 3(R vet r) 


prop. 50) = s(t + m) will represent the mass equivalent 


, which let be denoted by 3 the (Mechan. 


to the vical or lever when reduced to the point'a; while 
the bites equivalent to w, when referred to that point, 


will be — > —"w. ‘Hence, nae as In he Jast_ ‘prop. we 


2 
shall have a, (m-+m) pF Pew for the inertia to be over- 


mr ; | 
come ; and (p-+aM - nt ty a oe ~ (+m) af P age “ Wwe 
the accelerating force of p, or of w reduced to a. Mul- 
tiply this by w; and, for the sake of oe ti the pro- 


cess, put q for r ch in, and n for e ae =; (u -+ m), then 


y Nee Sem | 


then will” 72 — be a quantity wien varies as the effect 
varies, and which, indeed, when multiplied hi gt, denotes 
the effect itself. Putting the fluxion of this equal to nothing, 
and aa we at length find 


Be n? R? nr? 
=-y(t 2 v2 


Cor. When R=r, aie mM=m, if we restore the values of n 
and q, the expression will become w=,/(2P? cau 


(p-+2 2m). 4 
PROPOSITION Ill. 
Given the length land angle e of elevation of an inclined plane 

Bc; to fi nd the length u of another inclined plane ac along 
thick a given weight w shall be raised from the horizontal 
line ap to the point c, in the least tune possible, by means of 
another given weight p descending along the given plane cp : 
the two weights being connected by an imextensible thread ncw 
running always parallel to the two planes. 

Here we must as a preliminary 
to the solution of this proposition, 
deduce expressions for the motion 
of bodies connected by a thread, and 
running upon double inclined planes. 
‘Let the angle of elevation cap be © 
x, while e is the elevation cBp.’ 
Then at the end of the time 4, P 








422 _ MAXIMUM IN MACHINES. 


will have a velocity v; and gravity would impress upon it 
in the instant ¢ following, a new velocity =g sin e . 7, provid- 
ed the weight p were then entirely free : but, by the disposi- 
tion of the system, v will be the velocity which obtains in 
reality. Then, estimating the spaces in the direction cp, as 
the body w moves with an equal velocity but in a contrary 
sense, it is obvious, that by applying the 3d Law of Motion, 
the decomposition may be made as follows. At the end of the 
time t-++-7 we have, for the velocity impressed ‘on, | 


Vp weve effective veloc. frem c towards B. 
-v-+gsine. % where ; gsine.i—v,....-velecity destroyed. 
—v—~o. . effective ‘veloc. from c towards A. 


Wa 0 ig Site. 2,1hers 4 + g sin # . i,. . velocity destroyed. 


If, therefore, gravity impresses, during the time ¢ upon the | 
masses P, w, the respective velocities g since .# —wv, and g 
sing .¢-+v, the system will be in equilibrio.. The quanti- 
ties of motion being therefore equal, it will be 
pg sine .¢ —Pvu== wg sink .¢ + Wu. 
Whence the effective accelerating force is found, i. e. 
vy PSIn €—w SINE 
Pa han eal nee bata}, * Pe 
t p-+-w ce 
Thus it appears that the motion is uniformly varied, and we 
readily find the equations for the velocity and space from 
which the conditions of the motion are determined: viz. 
Pp sin e@—W SIDE Pp sin €—WSINE f 
ie Re eer teen d gtS 
p+w P-+w 
s(ptw) 


The latter of these two equations gives (?= 
38(F sine—w sine 


But in the triangle anc it is ac : Bc :: sin B : sin a, that is, 

















. * : e i =) 
u:l::sine:sine; hence— .=sin e, and — / = sin £; 
m m 


m being a constant quantity always. determinable from the 


_s@tw) 


— (PL — wi) 


data given. And 7? PES MOR cree Now when 


any ntianetey as t, is a Minimum, ite square is manifestly a mi- 
nimum : so that substituting for s its equal 1, and striking out 


= amin, or vats fluxion 





L2 
the constant factors, we have 
PL= Wil 


— ous 2 
Rit dee Whe EE = 0. Here: as in al sida cases, since 
~ (ep=wil) 

the fraction vanishes, its numerator must be equal to 0; con- 
sequently 2 QL? — 2win—pr2?=0, piu==2wil, ori: 1: : Qw : P; 

Cor. 1. Since neither sin e nor sin = enters the final equa- 
tion, it follows, that if the elevation of the plane sc is not giv- 
en, the problem is unlimited. Cor. 





MAXIMUM IN MACHINES. 423 


Cor. 2, When sin e=1, Bc coincides with the perpendicu- 
lar cp, and the power P acts with allits intensity upon the 
weight w. This is the case of the present problem which 
has it nso been considered. 


SCHOLIUM. . 


This proposition admits of a neat ' 
geometrical demonstration. Thus, 
let ce be the. plane upon which, if 
w were placed, it would be sustain- 
ed in equilibrio by the power P on 
the plane cp, or the power P hang- 
ing freely in the vertical cp; then 
(Mechan. prop. BB} «5 + 2 +e 
BC:cD:ceE:P:P’:w. Butwis to the force with which it 
tends to descend along the plane ca,as ca to cp; consequent- - 
ly, the weight p isto the same force in the same ratio; be- 
cause either of these weight in their respective positions 
would sustain w on CE, Therefore the excess of Pp above that 
force (which excess is the power accelerating the motions of 
Pp and w) is to Pp, as cA—ce to ca; or taking cH=ca, as BH to 
ca. Now, the spike being uniformly accelerated, we have 





SOFT; 0r T? K =; consequently, the square of the time Ne 


which ac’ is disecsibal by w, will be as ac directly, and as: ae 


2 


cA? 
inversely; ; and will be least when <n is a minimum ; that is, 


when — — +- En + 2cE, or (because 2ce is invariable) } when 


CE? 
sey + Enis a minimum. Now, as, when the sum of two 


quantities is given, their product isa maximum when they are 
equal to each other ; so it is manifest that when their product 
is given, their sum must be a minimum when they are equal. 


But the. product of = —— and EH is CE?, and consequently given ; 


therefore the sum ot vd ~ and EH Is least when those’ parts 2 are 


equal ; that i is when EH=CeE, Or eae Gini So that the length 
of the plane. ca is double the length of that on which the weight 
w would be kept in equilibrio by Pp acting along cB. _ 

When cp and ce coincide, the case becomes the same as 
_ that considered by Maclaurin, i in his View of Newton’s Philo- 


sophical Discoveries, pa. 183, Svo.-edit, 
PROPOSITION 


424 - MAXIMUM IN MACHINES. 


PROPOSITION IV. 
Let the given weight vp descend along ce, and by means of the 
thread pew (running parallel to the planes) draw a weight w 
up the plane ac: it’ 2s required to find the values of Ww, “when 
ats momentum is & maximum, the lengths and posttions of the 
planes being given. (See the preceding jig. we 
. sin €—wW sin E 
Pew 


1 
bgts ine by substitut. — Pa for sin e, and — a for sin E, becomes 
mm 


= (en —wil) | se Leemewed) 
gt. This mul. into w, gives-————_— Sig Ween 


The general expression for the vel. in ons 


vv == 








P-++w ! 
which, by the prop. is to be a maximum, | Or, striking out the 
PWL—Ww2l 

p--w 


this into fluxions, and reducing, we have p?.—2ewl- —w?/=0, 


or w=pr/ (4 +1)—p. 


Cor. When the inclinations of the planes are equal, 1, and 

i are equal, and w==P,/2—p=P X(,/2 — 1)='4142P : agree- 

ing with the conclusion of the lever’of equal arms, or the 

_ extreme ease of the wheel and axle, 1. e. the pulley. 

PROPOSITION V. 

Given the radius R of a wheel, and the radius r of its aale, the 
weight of both, w, and the distance of the centre of gyration 
from the axis of motion eg; also a given power P acting at the 
circumference of the wheel; to find the weight w raised by a. 
cord folding about the axle, so that its momentum shall be.c 
maximum. 

The force which absolutely: impels 
the point a is p, while w actsinadi- 
cea contrary to p, with a force tA 


=a max. Putting 








j 1 ; 
constant factors, —, gt, then is 
at 


; this therefore subducted from p, 


RP—Tw 
—, for the re- 
R 








rw 
leaves p—— = 
R 


5 ‘aes | 
duced force of. impelling the point 4. I - § OW 
And the inertia which resists the com- 
we aoe of staat to the point A will be the same as if the 


i were concentrated i in a point a (Mechan. 








mass sei 


prob. 50). If the former of these be divided by the latter, 
me R(RP-—?r —7r2 oti | 


sha betw w-+R?p 





757 is the force accelerating a : multi- 


plying 


the a 


MAXIMUM IN MACHINES. 42 


RrP—r2w | 
— for the f 
plying this by = , we have ——— dey ra wea: or the force which 
accelerates the weight w in its ascent. Consequently the ve- 
Rrep—r? Ww 


locity of w will be =—_ eww ——gt; which multiplied into 


RYPW—7r2 WwW 
tfor the momentum. As this is to be 
e2w-r2 w- +R? P CR? PO 


a maximum, its fluxion will = 0; whence we shall obtain w = 
of (Ri PPR -+ 2R2 Per w--e4w? +-pwrre? +P? R3r)—R? p—e? w 
CSPI ale Cs a So SO A a a 

Cor. 1. When R = 7, as in the Cue of the single fixed pul- 


e? 
ley, then w= 4/(2r9RI-+-2nKgtw-+ e Sw? -+ PwRe? )-Sje-r. 
Cor. 2. When the pulley is a esiiities of uniform matter 


e2==1n?, and the express, becomes w = ,/[R°(2P? gw + 
}u?)]—}w—p. 
Cor. 3. If, in the first general expression for the momen- 
Rrpw— 72 w?2 
aetrew 


=amaximum. Which, in fluxions and reduced, gives w = 
ae (q+Rrp) ~ =a. 
Cor. 4. If the moving force be destitute of inertia, then 
will ¢ = e?w and w, as in the last corollary. 
PROPOSITION VI. 
Let a given power P be applied to the cireumference of a wheel, 
ats radius R, to raise a weight w at its axle, whose radius is 
r, itis required to find the ratio of R and r he w is raised 


with the greatest momentum; the characters w and e denoting 
the same as in the last proposition. 


Ww gives 


tum of w, a be put = r2p-+e2w, we shall have 


Here we suppose r to vary in the expression for the mo- 

mentum of w, ——— rah als t. 
 o2wtriwtR te 
the conditions of any specified instance, we can ascertain what 
quantity of matter g shall make r?g = ¢?w, which, in fact, 
may always be done as soon as we can determine ¢. The ex- 
RrPw — 7? w2 
n?P+r2(q +w) Sew) 
The fluxion of which being made = 0, gives, after a ttle ree 
Ry [P2w?-+r8(9-+w)]—Pw 
EE), 

Cor. When the inertia of the machine is evanescent. with 

respect to that of P-+w, then is r=r 4/ (1+ -)=1, 


Von. If, 55 PROPOSITION 


And we suppose, that by 


pression for the work will then become —— 


- duction, r= 





426 MAXIMUM IN MACHINES. 


PROPOSITION VII. 


In any machine whose motion accelerates, the weight will be moved 
witle the greatest velocity, when the velocity of the power as to 


that of the weight, as 1+ pJ(it—) to 1; the inertia of the 


. e ¢ 
machine being disregarded. 
For any such machine may be considered as reduced to a 
lever, or to a wheel and axle Wes nay are R andr: in 


which the velocity of the weight — : seri wat (prop. 1) is to 


be a maximum, 7 being considered as variable. Hence then, 


following the usual rules, we find pR =r (w+.,/w?-+Pw). 

From which, since the velocities of the power and weight are 

respectively as R and r, the ratio in the proposition immedi- 

ately flows. 
Cor. When the weight moved is equal to the power, then is 

Re Pate Da Bos dy 3 ty Q4142 2) nearly. 

PROPOSITION VIII. 

If in any machine whose motion accelerates, the descent of one 
weight causes another to ascend, and the descending weight be 
given, the operation being supposed continually repeated, the 
effect will be greatest ina given tume when the ascending weight 

as to the descending weight, as 1 to 1:618, in the case of equal 
heights ; and in other cases, when it 1s te the exact counter- 
poise in a ratro which is always between 1 to 14 and 1 to 2. 


Let the space descended be 1, that ascended s; the de- 
scending weight 1, the ascending weight = then would the 


equilibrium require w = s ; and aatante will be the force act- 
w | 


ing on 1. Now the mass o reduced to the point at which 
Wa Oy ky | 2 
the mass 1 acts, will be = a 7 = ~- ; consequently the 


whole mass moved is equivalent to 1 -- —, and the relative 
! w 


, Gee s? ws 
force is Oe =(1 sae siete 


ing given, the time is as the root of the accelerating force 


w-4-s2 


inversely, that is, as 4/ ———-: andthe whole effect in a given 
j W=—$ 





. But, the space be- 


time, being directly as the weight raised, and inversely as 


, | . 1 w— f 
the time of ascent, will be as —,/ ea which must be a 
pl Rive a maximum. 


MAXIMUM IN MACHINES. 427 


maximum. Consequently its square ——— ae es oe — must be a max. 


likewise. ee latter expression, in Riis and reduced, 


gives w = — <1v(s +10s+9) —a-+-3]. 





Here ifs = 1, w= en nok : but if s be diminished with- 


out limit, w = 3s; if it Be augmented without limit, then will 
/ (s3 +10s-+9) approach indefinitely near to s+-5, and con-: 
sequently w =: 2s, Whence the truth of the Bronge ten is 
manifest. 


PROPOSITION IX. 


Let @ denote the absolute effort of any moving force, when it has 
no velocity ; and suppose it not capable of any effort when the 
velocity is w; let F be the effort answering to the ray v3 


then, if the force be uniform, F will be = 9 (1—- =). 


For it is the difference between the “blocitiee w and v 
which is efficient, and the action, being constant, will vary as 
the square of nt efficient velocity. Hence we shall have this 
analogy, @ : F :: We 0)? ; (w—v)? : consequently, r= 


w—v 
$( Thad -—). 

Though the pressure of an animal is not actually uniform 
during the whole time of its action, yet it is nearly so: so 
that in general we may adopt this hypothesis in order to ap- 
proximate to the true nature of animal action. On which 
supposition the preceding prop. as well as the remaining one, 
in this chapter will apply to animal exertion. 

Vv 

Cor. Retaining the same notation, we have w = 7, Tia 
This, applied to the motion of animals, gives this theorem: 
The utmost velocity with which an animal not impeded can move, 
ts to the velocity with which it moves when impeded by a given 
resistance, as the square root of its absolute force, to the differ- 
ence of the square roots of ats absolute and efficient forces. 


PROPOSITION X. 


To investigate expressions by means of rohan the maximum ef- 
fect in machines whose motion is uniform, may be determined. 





I. It follows from the observations made in art. 1 and the 
definitions in this chapter, that when a machine, whether 


simple or compound, is put into motion, the velocities of the 
impelled 


428 MAXIMUM IN MACHINES. 


impelled and working points, are inversely as the forces which 
are in equilibrio, when applied to those points in the direc-| 
tion of their motion. Consequently, if f denote the resistance 
when reduced to the working point, and v its velocity; while 
F and v denote the force acting at the impelled point, and its 
velocity ; we shall have rv =-fv, or introducing ¢ the time, 
Fvt=fvt. Hence, in all working machines which have acquir- 
ed a umform motion, the performance of the machine is equal to 
the momentum of impulse. 

II. Let F be the effort of a force on the impelled point of 
a machine when it moves with the velocity v, the velocity 
being w when Fr = 0, and let the relative velocity w—v=u. 
w 





—_ V . 
)?, the momentum of im- 


Then since (prop. 1x) F = $( 
U2 
w? 
v=w—wu. Making this expression for rv a maximum, 
or, suppressing the constant quantities, and making u?(w-—1w) 
a max. or its flux. = 0, when w is variable, we find 2w = 3u, 
oru =2w. Whence v = w—u=w—2w=1iw. 

Consequently, when the ratio of v to v is given, by the con- 
struction of the machine, and the resistance is susceptible of va- 
riation, we must load the machine more or less till the velocity of 
the umpelled point, is one-third of the greatest velocity of the 
force; then will the work done be a maximum. 

Or, the work done by an animal is greatest, when the velocity 
with which it moves, 1s one-third uf the greatest velocity with 
which wt is capable of moving when not impeded. 


; u? tw? 
II. Since r = pre mau acrers) = 49, in the case of the 


maximum .we have rv = 4¢9v = 492 w = ow, for the 
momentam of impulse, or for the work done, when the ma- 
chine is in its best state. Consequently, when the resistance 
1s @ given quantity, we must make v : v :: Of : 463; and this 
structure of the machine will give the maximum effect = ow. 
. IV. If we enquire the sreatest effect on the supposition 
that @ only is variable, we must make it infinite in the above 
expression for the work done, which would then become 


Vv Vv r 3 
WF, OF Woof Or Ww. ft, including the time in the formula. 


Hence we see, that the sum of the agents employed to move a 
machine may be infinite, while the effect is finite: for the varia- 
tions of 4, which are proportional to this sum, do not influ- 
ence the above expression for the effect. 


pulse rv will become v@ (—) ==. (w—w); because 


Scholiam. 


MAXIMUM IN MACHINES. 429 


Scholium. ? 


The propositions now delivered contain the most material 
principles in the theory of machines. The manner of apply- 
ing several of them is very obvious: the application of some, 
being less manifest, may be briefly illustrated, and the chapter 
concluded with two or three observations. 

The last theorem may be applied to the action of men and 
of horses with more accuracy than might at first be suppos- 
ed. Observations have been made on men and horses draw- 
ing a lighter along a canal, and working several days together. 
The force exerted was measured by the curvature and weight 
of the track-rope, and afterwards by a spring steelyard. » ‘The 
product of the force thus ascertained, into the velocity per hour, 
was considered asthe momentum. In this way the action of men 
was found to be very nearly as (wv)? the action of horses 
loaded so as not to be able to trot was nearly as (w—v) !'7, 


or as (w—v)>. Hence the hypothesis we have adopted may 
in many cases be safely assumed. 

According to the best observations, the force of a man at 
rest is on the average about 70 pounds ; and the utmost velo- 
city with which he can walk is about 6 feet per second, taken 
at a medium. Hence, in our theorems, ¢ = 70, and w = 6. 
Consequently r=4 =31 1 lbs. the greatest force a man can 
exert when in motion: and he will then move at the rate of 
tw, or 2 feet per second, or rather less than a mile and a half 
per hour. : 

The strength of a horse is generally reckoned about 6 times 
that of a man ; thatis, nearly 420lbs. at a dead pull. | His ut- 
most walking velocity is about 10 feet persecond. Therefore 
his maximum action will be 4 of 420= 1862 Ibs. and he will 
then move at the rate of 1 of 10, or 31 feet per second, or 
nearly 21 miles per hour. In both these instances we sup- 
pose the force to’ be exerted in drawing a weight along a ho- 
rizontal plane; or by raising a weight by acord running over 
‘a pulley, which makes its direction horizontal. 

2. The theorems just given may serve to show, in what 
points of view machines cought to be cousidered, by those who 
would labour beneficially for their improvement. 

The first object of the utility of machines consists in fur- 
nishing the means of giving to the moving force the most com- 
modius direction; and, when it can be done, of causing its 
action to be applied immediately to the body to be moved. 
These can rarely be united: but the former can be accom- 
plished in most instances ; of which the use of the simple 

lever 


430 MAXIMUM IN MACHINES. 


lever, pulley, and wheel and axle, furnish many examples. 
The second object gained by the use of machines, 1s an ac- 
commodation of the velocity of the work to be performed, to the 
velocity with which alone a natural power can act. Thus when- 
ever the natural power acts with a certain velocity which can- 
not be changed, and the work must be performed witha great- 
er velocity, a machine is interposed moveable round a fixed 
support, and the distances of the impelled and working points 
are taken in the proportion of the two given velocities. 


But the essential advantage of machines, that, in fact, which 
properly appertains to the theory of mechanics, consists in 
augmenting, or rather in modifying, the energy of the mov- 
ing power, in such manner that it may produce effects of 
which it would have been otherwise incapable. ‘Thus a man 
might carry upa flight of steps 20 pieces of stone, each 
weighing 30 pounds (one by one) in as small a time as he could 
(with the same labour) raise them all together by a piece of 
machinery, that would have the velocities of the impelled and 
working points as 20 to 1; and in this case, the instrument 
would furnish no real advantage, except that of saving hig 
steps. Butif a large block of 20 times 30, or 600lbs. weight 
were to be raised to the same height, it would far surpass the 
utmost efforts of the man, without the intervention of some 
such contrivance. 


The same purpose may be illustrated somewhat differently ; 
confining the attention all along to machines whose motion is 
uniform. The product fv represents, during the unit of time, 
the effect which results from the motion of the resistance ; 
this motion being produced in any manner whatever. If it be 
produced by applying the meving force immediately to the 
resistance, it is necessary not only that the products rv and fv 
should be equal; but that at the same time F=f, and v=v: 
if, therefore, as most frequently happens, f be greater than Fr, - 
it will be absolutely impossible to pat the resistance in motion 
by applying the moving force immediately to it.. Now ma- 
chines furnish the means of disposing the product rv in such 
a manner that it may always be equal to fv, however much 
the factors of rv may differ from the analogous factors in fv ; 
and, consequently, of putting the system in motion, whatever 
is the excess of f over F. 


Or, generally, as M. Prony remarks (Archi. Hydraul. art. 
504), machines enable us to dispose the factors of rvt in such 
a manner, that while that product continues the same, its fae- 
tors may have to each other any ratio we desire. | If, for in- 
stance, time be precious, the effect must be produced ina very 

short 


MAXIMUM IM MACHINES. 431 


short time and yet we should have at command a force capa- 
ble of little velocity but of great effort, a machine must be 
found to supply the velocity necessary for the intensity of the 
force: if, on the contrary, the mechanist has only a weak pow- 
er at his disposition, but capable of a great velocity, a machine 
must be adopted that will compensate, by the velocity the 
agent can communicate to it, for the force wanted : lastly, if 
the agent is capable neither of great effort, nor of great velo- 
city, a convenient machine may still enable him to accomplish 
the effect desired, and make the product rvt of force, veloci- 
ty and time, as great as is requisite. Thus, to give another 
example: Suppose that a man exerting his strength imme- 
diately on a mass of 25 Ibs, can raise it vertically with a velo- 
city of 4 feet per second; the same man acting on a mass of 
1000 lbs. cannot give it any vertical motion though he exerts 
his utmost strength unless be has recourse to some machine. 
Now he is capable of producing an effect equal to 25 X 4 X 
t: the letter ¢ being introduced because, if the labour is con- 
tinued the value of ¢ will not be indefinite, but comprised 
within assignable limits. Thus we have 25 X 4 X t= 1000 
Xvxt; and consequently v = ~, of a foot. This man may 
therefore with a machine, as a lever, or axis in peritrochio, 
cause a mass of 1000 lbs to raise 5, of a foot, in the same time 
that he could raise 25 lbs. 4 feet without:a machine ; or he 
may raise the greater weight as far as the less, by employing 
AU times as much time. : , 

From what has been said on the extent of the effects which 
may be attained by machines, it will be seen that, so long as 
a moving force exercises a determinate effort, with a velocity 
also determinate, or so long as the product of these is con- 
stant, the effect of the machine will remain the same : thus, 
under this point of view, supposing the preponderance of the 
effort of the moving power, and abstracting from inertia and 
friction of materials, the convenience of application, &c. all 
machines are equally perfect. But from what has been 
shown, (props. 9, 10) a moving force may, by diminishing 
its velocity, augment its effort and reciprocally. There is 
therefore a certain effort, of the moving force, such that its 
product by the velocity which comports to that effort, is the 
greatest possible. Admitting the truth of the law assumed 
in the propositions just referred to, we have, when the effect 
is a maximum, v = 1w, or F = 4; and these two values 
. obtaining together, their product 4, gw expresses the value 
of the greatest effect with respect to the unit of time. In 
practice it will always be adviseable to approach as nearly to 
these values as circumstances will admit; for it cannot be 

expected 


432 PRESSURE OF EARTH AND FLUIDS. 


expected that they can always be exactly attained. But a 
small variation will not be of much consequence: for, by a 
well known property of those quantities which admit of a 
proper maximum and minimum, a value assumed at a mode- 
vate distance from either of these extremes will produce no 
sensible change in. the effect. | 

If the relation of ¥ to v followed any other law than that 
which we have-assumed, we should find from the expression’ 
of that law values of r, v, &c. different from the preceding. 
The general method however would be nearly the same. 

With respect to practice, the grand object in all cases 
should be to procure a uniform motion, because it is that from 
which (ceteris paribus) the greatest effect always results. 
Every irregularity in the motion wastes some of the impelling 
‘power: and it is the greatest only of the varying velocities 
which is equal to that which the machine would acquire if it 
moved uniformly throughout: for, while the motion accele- 
rates, the impelling force is greater than what halances the re- 
sistance at that time opposed to it, and the velocity is less than 
.what the machine would acquire if moving uniformly; and 
when the machine attains its greatest velocity, it attains it be- 
cause the power is not then acting against the whole resistance. 
In both these situations therefore, the performance of the ma- 
chine is less than if the power and resistance were exactly 
balanced ; in which case it would move uniformly (art. 1.) 
Besides this, when the motion of a machine, and particularly 
"avery ponderous one, is irregular. there are continual repeti- 
tions of strains and jolts which soon derange and ultimately 
destroy the whole structure. Every attention should there- 
fore be paid to the removal of all causes of irregularity. 


ail 


PRESSURE OF EARTH AND FLUIDS AGAINST WALLS 
AND FORTIFICATIONS, THEORY OF MAGAZINES, 
&c. 


PROBLEM I. 
To determine the Pressure of Earth against Walls. 


Wuen new-made earth, such as is used in forming ram- 
parts, &c. is not supported by a wall as a facing, or by coun- 
terforts and land-ties, &c. but left to the action of its weight 
and the weather ; the particles loosen and separate frem each 

other, 


“AGAINST WALLS, &c. 433 


other, and form a sloping surface, nearly regular; which 
plane surface is called the natural slope of the earth ; and is 
supposed to have always the same inclination or deviation 
from the perpendicular, in the same kind of soil. In com- 
mon earth or mould, being a mixture of all sorts thrown to- 
gether, the natural slope is commonly at about half a right 
angle, or 45 degrees ; but clay and stiff loam stands at a greater 
angle above the horizon, while sand and light mould will only 
stand at a much less angle. The engineer or builder must 
therefore adapt his calculations accordingly. 

Now, we have already given, (at prop. 45 Statics) the 
general theory and determination of the force with which 
the triangle of earth (which would slip down if not sup- 
ported) presses against the wallonthe ¢ BR AG 
most unexceptional principles, acting eine | 
perpendicularly against ar at x, or 1 of 
the altitude az above the foundation at 
E; the expression for which force was 


3 
there found to Bem 





B2 
—m; where’ m ~- 
OBE? 


denotes the specific gravity of the earth of the triangle ane.— 
It may be remarked that this was deduced from using the area 
only of the profile, or transverse triangular section-aBE, in- 
stead of the prismatic solid of. any given length, having that 
triangle for its base. And the same thing is done in determin- 
ing the power of the wall to support the earth, viz. using only 
its profile or transverse section in the same plane or direction 
as the triangle ane. This it is evident will produce the same 
result as the solids themselves, since, being both of the same 
given length, these have the same ratio as their transverse 
sections. 

In addition to this determination, we may here further ob- 
serve that this pressure ought to diminish in proportion to the 
cohesion of the matter in sliding down the inclined plane se. 
Now it has been found by experiments, that a body requires 
about one-third of its weight to move it along a plane surface. 
The above expression must therefore be reduced in the ratio 


AE? . AB? 
—-m for the true 
9RE2 








of 3 to 2; by which means it becomes 
practical efficacious pressure of the earth against the wall. 
. AB F e é 2 
Since—, which occurs in this expression of the force of 


the earth, is equal to the sine of the Z ars to the radius 1, 
put the sine of that 7 x =e; also put a= ae the altitude 
ofthe triangle ; then the above expression of the force, viz. 

Vou. Il. 56 AES , AB? 


434 PRESSURE OF EARTH AND FLUIDS 


AES . AB? why 
oa te becomes ta3c?m, for the perpendicular pressure 
BE 7 


of the earth against the wall. And if that angle be 45°, as is 
usually the case in common earth, then is e? = 3, and the 
pressure becomes ;';a?m. 





PROBLEM II. 
To determine the Thickness of Wall to eRe the Earth. 
In the first place suppose the section Sane, Ae 


of the wall to be a rectangle, or equally. iD es 
thick at top and bottom, and of the same- ie AG 
height as the rampart of earth, like arre Ds KI 

in the annexed figure. Conceive re i iD | 
weight w, proportional to the area cz, E 3 
to be appended to the base directly be- Ww" 

low the centre of gravity of the figure. Now the pressure of 
the earth determined in the first problem, being in a direction 
parallel to ac, to cause the wall to overset and turn back 
about the point r, the effort of the wall to oppose that effect, 
will be the weight w drawn into rn, the length of the levee 
by which it acts, that is w X FN, or aEFG X FN in general, 
whatever be the figure of the wall. 

But now in case of the rectangular figure, the area GE=AE 
XEF =az, putting a = akg the altitude as before, and x = EF 
the required thickness ; also in this case rn = ter = iz, the 
centre of gravity being in the middle of the rectangle. Hence 
then ax X ta =1ax?, or rather 1az2n is the effort of the 
wall to prevent its being overturned, n denoting the specific 
gravity of the wall. 

Now to make this effort a due balance to the pressure of 
the earth, we put the two opposing forces equal that, is 
Lax?n = 1a3e2m, or 142n = 1a3e2m, an equation which gives 













————— 





2 re : ; 
r= hae /—, for the requisite thickness of the wall, just to 


sustain it in equilibrio. 
Corol. 1. The factor ae, in this expression, is = the line 
aQ drawn perp. to the slope of earth pe: theref. the breadth 


2 : nee 
x becomes = 1a@ 1/ — , which conseq. is directly propor- 


tional to the perp. AQ. When the angle at © is = 45°, or 
_ halfa right angle, as is commonly the ah its sine eis = ,/4, 


and the breadth of the wall 2 = hay — Further, when the 


wall is of brick, its specific gravity is Sia) the same as 
the earth, or m=n, and then its thickness x=14a, or one-third 
of its height. —But when the-wall is of stone, of the specific 

gravity 


AGAINST WALLS, &c. — ert 


gravity 21, that of earth being nearly 2, that is, m = 2, and 
n=21; then /—= fi 895, 1 of which is -298, and the 


breadth r=: 298u= #54 nearly. Thats is, the thickness of the 
stone wall must be ,3; - of ‘its height. 


PROBLEM III. 


To determine the Thickness of the Wall at the Bottom, when 
ats Section is a Triangle, or coming to an Edge at Top. 

In this case, the area of the fail AEF 
is oo half of what it was before, or 
only Lam X er =1az, and the weight 
w=ilaen. But now, the centreof gra- 
vity is at aly of re from the line 4x, 
or FN = 2re = 2a. Consequently rm 
Xw=2 2x Xlarn==1ax? n. This, as be- 
fore, being. put = the pressure of the 
earth, gives the equation 1ax2n = 1a%e2m, or x7n == 107e? m, 





and the root x, or thickness rr = ae ,/ = =a./ = for the 
; n 27. 


slope of 45°. Ces 
Now when the wall is of brick, oF m =n nearly, this be- 

comes « = a,/2= 4080 = 2a, or ;4, of the height nearly. 
But when the wall is of stone, Gn. a nas 2 to 21, then 


J f =4, and the thickness x or a J =a % = °365a 


=a nearly, or nearly 2 of the height, 
PROBLEM IV. 


To determine the Thickness of the Wull at the top, when the Face 
1s not Perpendicular, but Inclined as the Front of a Fortifica- 
tion Wall usually is. 

Here cr represents the outer face of HU ASC 
a fort, aere the profile. of the wall, hav- ae Rae 
ing ac the thickness at top, and er that \ ‘i i“ » i) 
at the bottom. Draw cu prep. to EF ;_ ji a ih 
and conceive the two weights w, w, to be AAT HN 
suspended from the centres of gravity 
of the rectangle au and the triangle cur, 
and to be proportional to their areas 
respectively. Then the two momenta of the weights W, @, 
acting by the levers rn, rm, must be made equal to the pres- 
sure of the earth in the direction prep. to az. 

Now put the required thickness ac or EH =<, and the 
altitude aE or Go =a as before. And,-because in such cases 
the slope of the wall is usually made equal to 4 of its alhinde 

that 















436 _ PRESSURE OF EARTH AND FLUIDS 


that is FH =: 1 ak or 1a, the lever rm willbe 2 of 1a = ja, 
and the leer FN = FH + AnH =la-+ is. * But the area 
of cor ==cH X lHF =aX jj, a= j, a? = w, and the area 
AH = aE X aG = a4 = Ww; ‘these two drawn into the re- 
spective levers rm, FN, give the two momenta, 2,aw = ,2.aX 
joo? = 7;a?, and (ta-+ 42) X az = 102x + Laz? ; theref. 
the sum of the two, (sex? +1a?2-+7 an must be = ;;a°m 


or dividing by jan, x? -- Zax +- 7,09 = ta? X = ; now add- 
ing a2 to both sides to comptese: the square, the equation 


becomes a? + 2ax-+ 54 a? =1a? + ha? the root of which 


is a+la=a,/ (45 Hhige =): and hence « = a ,/ (3,+ —)- ta. 
And the base Er = a 7 (as + =). ; . 


Now, for a brick wall, m =n nearly, and then the breadth 
e=aV/S (q+ 1)—la = T34/f 34 — ta = 1894, or almost 


1a in brick walls.—But the stone walls, a= 4,andz=a,/ 
n 


(dstay) — Fo=pay/29—1a = +1590 = 54,0 nearly, for the 
thickness 4c at the top, in stone walls, 

In the same manner we may proceed when the rape! 
supposed to be any other part of the altitude, instead of 4 
usedabove. Ora ai solution might be given, by Heb 


ing the thickness oi part of the altitude. 


REMARK. 


Thus then we have given all the calculations that may be 
necessary in determining the thickness of a wall, proper to 
support the rampart or body of earth, in any work. If it 
should be objected, that our determination gives only such a 
thickness of wall, as makes it an exact mechanical balance to 
the pressure or push of the earth, instead of giving the 
former a decided preponderance over the latter, as a security 
against any failure or accidents. ‘To this we answer, that 
what has been done is sufficient to insure stability, for the 
following reasons and circumstances. First, it is usual to 
‘build several counterforts of masonry, behind and against the 
wall, at certain distances or intervals from one another ; which 
contribute very much to strengthen the wall, and to resist the 
pressure of the rampart. 2dly. We have omitted to include 
the effect of the parapet raised above the wall ; which must 
add somewhat, by its weight, to the force or resistance of the 
wall. 


AGAINST WALLS, &c. 437 


wall. It is true we could have brought these two auxiliaries 
to exact calculation, as easily as we have done for the wall 
itself: but we have thought it as well to leave these two ap- 
pendages, thrown in as indeterminate additions, above the 
exact balance of the wall as before determined, to give it an 
assured stability. Besides these advantages in the wall itself, 
certain contrivances are also usually employed to diminish the 
pressure of the earth against it : suchas land-ties and branches, 
laid in the earth, to diminish its force and push against the 
wall. For ali these reasons then, we think the practice of 
making the wall of the thickness as assigned by our theory, 
may be safely depended on, and profitably adopted ; as the 
additional circumstances, just mentioned, will sufficiently in- 
sure stability ; and its expense will be less than is incurred 
by any former theory. : 


PROBLEM V. 


To Determine the Quantity of Pressure sustained by a Dam or 
Sluice, made to pen up a Body of Water. 


By art. 313 Hydrostatics, (in this volume) the pressure of 
a fluid against any upright surface, as the gate of a sluice or 
canal, is equal to half the weight of a column of the fluid, 
whose ‘base is equal to the surface ‘pressed, and its altitude 
the same as that of the surface. Or, by art. 314 of the same, 
the pressure is equal to the weight of a column of the fluid, 
whose base is equal to the surface pressed, and its altitude 
equal to the depth of the centre of gravity below the top or 
surface of the water; which comes to the same thing as the 
former article, when the: surface pressed is a rectangle, be- 
cause its centre of gravity is at half the depth. 

Ex. 1. Suppose the dam or sluice be a rectangle whose 
length, or breadth of the canal, is 20 feet, and the depth of 
water 6 feet. Here 20X6=120 feet, is the area of the sur- 
face pressed ; and the depth of the centre of gravity being 3 
fegt, viz. at the middle of the rectangle ; therefore 120x3= 
360 cubic feet is the content of the column of water. But 
each cubic foot of water weighs 1000 ounces, or 621 pounds ; 
therefore 360 X 1000 = 360000 ounces, 22500 pounds, or 10 
tons and 100ib. is the weight-of the éoltnnt of water, or the 
quantity of pressure on the gate or dam. 

Ex. 2. Suppose the breadth of a canal at the top, or sur- 
face of the water, to be 24 feet, but at the bottom only 16 
feet, the depth of water being 6 feet, as in the last example : 
required the pressure on'a gate which, standing across the 
canal, dams the water up? | 
Here 


438 PRESSURE OF EARTH AND FLUIDS 


Here the gate is in form of a trapezoid, 
having the two parallel sides aB, cp, viz. aB 
=24, and co=16, and depth 6 feet. Do; 
by mensuration problem 3, volume J,: 
(as-+cp) x 6, = 20K 6 = 120 the area of 
the sluice, he same as before in-the Ist ex- 
ample: but the centre of gravity cannot be 
so low down as before, because the figure is 
wider above and narrower below, the whole 
depth being the same. 

Now, to determine the centre of gravity 
K of the trapezoids ap, produce the two sides 
Ac, BD, till they meet in gc ; a draw GKeE and 
cH perp. toaB: then an: cH:: ak: Ge, thatis,4:6:: 12: 
18=ce; and EF being =6, theres: FG=12, Now, by Statics 
art. 229, er—=6=1kc gives F the centre of gravity of the tri- 
angle apc, and ri=4= 3¥Fa gives 1 the centre of gravity of the 
triangle coc. Then assuming x to denote the centre of an, 
it will be, by art. 212 this vol. as the trap. ap: A cba ::1F: 
FK, or A aBC — A CDG: A cbG@::1F : FK, or by theor. 88 
Geom. Ge? —GF? : GF? :: 1f : FK, thatis 182 —122 to 12? or 
3? —2? to 22 or 5:4: :1F=4: §=31=px; and hence ex 
==6 — 31 = 24 = 14 1s the distance of the centre x below the 
surface of the water. This drawn into 120 the area of the 
dam-gate, gives 336 cubic feet of water = the pressure, = 
336000 ounces = 21000 pounds = 9 tons 801b. the quantity 
of pressure against the gate, as required, being a 15th part less 
than in the first case. 

Ex. 3. Find the quantity of pressure against a dam or 
sluice, across a canal, which is 20 feet wide at top, 14 at bot- 
tom, and 8 feet depth of water ? 


PROBLEM VI. 


49, determine the Strongest Angle of Position of a Pair of Gates 
for the Lock on a Canal or River. 


Let ac, sc be the two gates, meet- D 
ing in the angle c, projecting out — HS opr a 
against the pressure of the water, aB 
being the breadth of the canal or river. 
Now the pressure of the water ona 

gate ac, is as the quantity, or as the A E % 
extent or length of it, ac. And the mechanical effect of that 
pressure, is as the length of lever to the middle of ac, or as 
ac itself, On both these accounts then the pressure is as 
ac*. Therefore the resistance or the strength of the gate 
must be as the reciprocal of this ac?. 








Now 


AGAINST WALLS, &c. 439 


Now produce ac to meet ep, prep. to it, inp; and draw 
cE to bisect aB perpendicularly in 2; then, by similar trian- 
gles, as AC: AE :: AB : AD; where, ag and aB being given 
lengths, ap is reciprocally as ac, or ap? reciprocally as ac? ; 
that is, ap? is as the resistance of the gate ac. But the re- 
sistance of ac is increased by the pressure of the other gate 
in the direction Bc*. Now the force in sc is resolved into the 
two BD, vc; the latter of which, pc, being paraliel to ac, has 
no effect.upon it; but the former, sp, acts perpendicularly on 
it. Therefore the whole effective strength or resistance of 
the gate is as the product ap? X Bp. 

If now there be put an =a, and sp =z, then ap? = ap? 
—BD? = a? — x? ; conseq. AD? XBp=(A? — 2?) X2=a2a — “3 
for the resistance of either gate. And, if we would have this 
to be the greatest, or the resistance a maximum, its fluxion 
_must vanish, or be equal to nothing : that is. a?2—3x?7=0 : 
hence a? =32?, and « = a,/1=1a,/3==-57735a, the natural 
sine of 35° 16’: that is, the. strongest position for the lock 
gates, is when they make the angle a or B = 35° 16’, or the 
compleniental angle acm or pce = 54° 44’, or the whole sa- 
lient angle acs = 109° 28’, 


Scholium. 


Allied to this problem, are several other cases in mechan- 
ics, such as, the action of the water on the rudder of a ship, in 
sailing, to turn the ship about, to alter her course ; and the 
action of the wind on a ship’s sails, to impel her forward ; 
also the action of water on the wheels of water-mills, and Ms 
the air on the sails of wind-mills, to cause them to turn round. 


Thus, for instance, let 
apc be the rudder of a 
ship ABDE, Sailing in the 
direction Bp, the rudder 
placed in the oblique po- 
sition Bc, and consequent- es 
ly striking the waterinthe © & bp ey 
direction cr, parallel to sv. Draw er prep. to Bc, and Bc 
prep. to cr. Then the sive of the angle of incidence, of the 
direction of the stroke of the radder against the water, wil! 
be gr, to the radius cr ; therefore the force of the water 
against the rudder will be as er”, by art. 3, Mot. of bod. in 
Flui. this vol. But the force nr resolves into the two BG, CF, 
of which the latter is parallel to the ship’s motion, and there- 











———— 


* The reasoning in this and in two following sentences does not appear to mé 
to be founded on mechanical principles. 


fore 


440 PRESSURE OF EARTH AND. FLUIDS 


fore has no effect to change it ; but the former se, being prep. 
to the ship’s motion, is the only part of the force to turn the 
ship about and change her course. But pr: BG :: cF : CB, 


» BC. BR? 
therefore cr : CB :: BF? : - —the force upon the rudder 





to turn the ship about. | 
Now put a = cr, = Bc; then pr? = a? — g?, and the 
2 2 2 27 —~ 973 : 
Ps lee Gea a and, to have this a 
maximum, its flux. must be made te vanish, that is, a?7—3a? 
x= 0; and hence « =a,/1= sc = the natural sine of 35° 
16’= angle r; therefore the complemental angle c= 54°44’ 
as before, for the obliquity of the rudder, when it is most 
efficacious. | 

The case will be also the same with respect to the wind 
acting on the sails of a wind-miil, or of a ship, viz. that the 
sails must be set so as to. make an angle of 54° 44’ with the 
direction of the wind; at least at the beginning of the motion, 
er nearly so when the velocity of the sail is but small in com- 
parison with that of the wind ; but when the former is pretty 
considerable in respect of the latter, then the angle ought to 
be proportionally greater, to have the best effect, as shown in 
Maclaurin’s Fluxions, pa. 734, &c. 3 

A consideration somewhat related to the same also, is the 
greatest effect produced on a mill-wheel, by a stream of water 
striking upon its sails or float-boards. ‘The proper way in 
this case seems to be, to consider the whole of the water as 
acting on the wheel but striking it only with the relative ye- 
locity, or the velocity with which the water overtakes and 
strikes upon the wheel in motion, or the difference between 
the velocities of the wheel and the stream. This then is the 
power or force of the water: which multiplied by the velo- 
city of the wheel, the product of the two, viz. of the relative 
velocity and the absolute velocity of the wheel, that is (v—v) 
v=vv—v?2, will be the effect of the wheel; where v denotes 
the given velocity of the water, and v the required velocity 
of the wheel. Now, to make the effect vv — v2 a maximum, 
or the greatest, its fluxion must vanish, that is v~y — 2vv = 0, 
hence v = 1v; or the velocity of the wheel will be equal to 
half the velocity of the stream, when the effect is the greatest ; 
and this agrees best with experiments. 

A former way of resolving this problem was, to consider 
the water as striking the wheel with a force as the square of 
the relative velocity, and this multiplied by the velocity of 
the wheel, to give the effect; that is, (v—v)*v = the effect. 
Now the flux. of this product is (v—v)?u—(v—v) X 2vv =0; 

: hence 





force 


AGAINST WALLS, &c. 441. 


hence v — v = 2v, or v= 3v, and v = tv, or the velocity of 
the wheel equal only to 1 of the velocity of the water. 
PROBLEM VII. 


To determine the Form and Dimensions of Gunpowder Maga- 
: zines. . 


In the practice of engineering, with respect to the erection 
of powder magazines, the exterior shape is usually made like 
the roof of a house, having two sloping sides, forming two in- 
clined planes, to throw off the rain, and meeting in an angle 
or ridge at the top; while the interior represents a vault, 
more or less extended, as the occasion may require ; and the 
shape or transverse section, in the form of some arch, both 
for strength and commodious room, for placing the powder 
barrels. It has been usual to make this interior curve a se- 
micircle. But, against this shape, for such a purpose, I must 
enter my decided protest : as it is an arch the farthest of any 
from being in equilibrium in itself, and the weakest of any, 
by being unavoidably much thinner in one part than in others. 
Besides it is constantly found, that after the centering of se- 
micireular arches is struck, and removed, they settle at the 
crown, and rise up at the flanks, even with a straight horizon- 
tal form at top, and still much more so in powder magazines 
with a sloping reof; which effects are exactly what might be 
expected froma contemplation of the true theory of arches. 
Now this shrinking of the arches must be attended with other 
additional bad effects, by breaking the texture of the cement, 
after it has been in some degree dried, and also by opening the 
joints of the voussoirs at one end. Instead of the circular 
arch therefore, we shall in this place give an investigation, 
foundéd on the true principles of equilibrium, of the only 
just form of the interior, which is properly adopted to the 
usual sloped roof. 

For this purpose, put @ =pk the’ 
thickness of the arch at the top, « 
= any absciss pp of the required 
arch apcm, u =«R the correspond- 
ing absciss of the given exterior 
line x1, and y=rc = rt their equal 
ordinates. Then by the principles 
of arches, in my tracts on that sub- 
ject it is found that cr or w= a+a 


—u=qx , or =Q x supposing y a constant quan- 








tity, and where @ is some certain quantity to be determined 
hereafter. Butxr or wis = ty, if t be put to denote the tan- 
Von. Il. 5'7 gent 


442 PRESSURE OF EARTH AND FLUIDS 


gent of the given angle of elevation xin, to radius 1, and then 
if ie ° _ Qa in fel 

the equation is pene Me 91 
Now, the fluxion of the equation 
w=a--r— ty, is Wx —ty and the 
2d fluxion is w= x; therefore the 
foregoing general equation becomes 


Qw Bo QED 
w= -—; and hence wyw=—— , the 
y* | oe 





ay 
fluent of which gives gt : but 
at p the value of w is =a, and w=0, the curve at p being pa- 
rallel to x1; therefore the correct fluent is w? — a? = —_. 
Qwes , eee 
72 a2 BS aaa a/ (w? —a a?)’ 
2g? 
fluent of which gives y = 4/@ X hyp. log. of —— iis Ssh : ee) 


Hence then y? = ; the correct 


Now, to determine the value of @, we are to FAO SV that 
when the vertical line ci is in the position at or un then w=cr 
becomes =at or Mn = the given quantity ¢ suppose, and y= 
AQ or gam=b suppose, in which position the last equation be- 


Hleiod), 


comes b =,/@ X hyp. log. —~¥ ; and hence it is 


found that the value of the constant adantiee af RISME ee 


ae Ls pater har) ; which being substituted for it, in the 


above seen value of y, that value becomes 














_ Tog. of wy (w? —a? ) 
apap soy a ie log. of w+ y/(w®—a?)—log. a 
Ue los ores t+y/ ee log. eae —a?)—log.a 


from which equation the value of the ordinate pc may always 
be found, to every given value of the vertical cr. 

But if, on the other hand, Pe be given, to find cr, which 
will be the more convenient way, it may be eae in the 


following manner: Put a = log. of a, andc == X log. of 
ae =) 

; then the above equation gives cy +- a = log. 
of w oN VW (w? ey again, put n = the number whose log. 
is cy A; then n =w + v (w? — a?); and hence w = 





on a | Now, 


_ AGAINST WALLS, &c. 443 


Now, for an example in numbers, ina real case of this 
nature, let the foregoing figure represent a transverse verti- 
cal section of a magazine arch balanced. in all its parts, in 
which the span or width am is 20 feet, the pitch or height pq 
is 10 feet, thickness at the crown pk = 7 feet, and the angle 
of the ridge tks 112° 87’, or the half of it Lxp = 56° 184, 
the complement of. which, or the elevation xir, is 33° 41, 
the tangent of which is = 3, which will therefore be the 
value of ¢ in the foregoing investigation. The values of the 
other letters will be as follows, viz. DK==a=7: ag==B=10; 
1S a 10; au =c = 401 =%; a = log. of 7 = 8450980 ; 
om : X log. of stan log. o Uns 8S = s 
log. of 2°56207 == -0408591; cy + a = -O408591ly + 
-8450980 = log. of nm. From the general equation then, viz. 

a2-+n?  q? Dey. : ‘al 
On on 1 8M by assuming y successively 
equal to 1, 2, 3, 4, &c. thence finding 
the corresponding values of cy -+- @ or 
-0408591y + -8450980, and to these, 
as common logs. taking out the corres- 


cl = Ww = 





Val. of y|Val. of w 
or cr. | or ct. 


—|— 














1 | 7-0309 
ponding natural numbers, which will 9 751943 
be the values of n; then the above 3 7:9806 
theorem will give the several values 4 75015 
of w or cr, as they are here arranged 5 77888 
in the annexed table, from which the | 6 8-14.52 
figure of the curve is to be construct- " 8:5737 
ed, by thus finding so many points in it. 8 90781 
Otherwise. Instead of making n the 9 9:G628 
number of the log. cy + a, if we put | 10 . 110°3333 
m = the natural number of the log. rats 
2 a2 
Meads, 7 sand am — w= 4/ (w? — 


cy only ; then m= 

a*), or by squaring, &c. Raa ite: + w? == w? — a?, and 

hence w= — x a: to which the numbers being applied, 
m 


fe very same conclusions result as in the foregoing calcula- 
tion and table. 


PROBLEM VIII. 
To construct Powder Magazines with a Parabolical Arch. 


It has been shown, in my tract on the Principles of Arches 
of Bridges, that a parabolic arch is an arch of equilibration, 
when its extrados; or form of its exterior covering, is the 

very 


ev THEORY AND PRACTICE 


‘very same parabola as the lower or inside curve. Hence then 
a parabolic arch, both for the inside and outer form, will be 
very proper for the structure of a powder magazine. For, 
the inside parabolic shape will be very convenient as to room 
for stowage : 2dly, the exterior parabola, every where paral- 
lel to the inner one, will be proper enough to carry off the 
rain water: 3dly, the structure will be in perfect equilibrium: 
and 4thly, the parabolic curve is may constructed, and the 
structure erected. 

Put, as before, a = kp,h = ve, 
Seana = pp, and y = rc or RI. 
Then, by the nature of the para- 


bola apc, 6? de ea Pe Ue: ey 


-  thyy 
h = 
ence x bb 
Seale flame Se Pes. 
by making y constant. Then cr = ax Q is = te == a con- 
stant quantity =a, what it is at the vertax ; that is, cr is every 
where equal to xp. 

Consequently KR is = pp; and since RI is = ec, it 1s evi- 
dent that x1 is the same parabolic curve with pc, and may 
be placed any height above it, always producing an arch of 
equilibration, and very commodious for powder magazines. 











——=— ge 


THEORY AND PRACTICE OF GUNNERY. 


In the Doctrine of Motion, Forces, &c. have been given 
several particulars relating to this subject. Thus, in props. 
19, 20, 21, 22, is given all that relates to the parabolic theory 
of projectiles, that is, the mathematical principles which 
would take place and regulate such projects if they were not 
impeded and disturbed in their motions by the air in which 
they move. But from the enormous resistance of that me- 
dium, it happens, that many military projectiles, especially 
the smaller balls discharged with the higher velocities, do 
not range so far as a 20th part of what they would naturally 
do in empty space! That theory therefore can only be use- 
ful in some few c. es, such as in the slower kind of motions, 
not above the velocities of 2, 3, or 400 feet per second, when 
the path of the projectile differs but little perhaps from the 
curve of a parabola. 

Again, at art. 104, &c. of same doctrine, are given several 
other practical rules and calculations, depending partly on the 

fores 


_- OF GUNNERY. A445 


foregoing parabolic theory, and partly on the results of cer- 
{ain experiments performed with cannon balls. 

Again, in prop. 58, Statics, are delivered the theory and 
calculations of a beautiful military experiment, invented by 
Mr. Robins, for determining the true degree of velocity with 
which balls: are projected from guns, with any. charges of 
powder. The idea of this experiment, is simply, that the 
ball is discharged into a very large but moveable block of 
wood, whose small velocity, in consequence of that blow, can 
be easily observed and accurately measured. Then, from 
this small velocity, thus obtained, the great one of the ball is 
immediately derived by this simple proportion, viz. as the 
weight of the ball, is to the sum of the weights of the ball 
and the block, so is the observed velocity of the last, to a 4th 
proportional, which is the velocity of the ball sought.—It is 
- evident that this simple mode of experiment will be the source 
of numerous useful principles as results derived from the 
experiments thus made, with all lengths and sizes of guns, 
with all kinds and sizes of balls and other shot, and with all 
the various sorts and quantities of gunpowder ; in short, the 
experiment will supply answers to all inquiries in projectiles, 
excepting the extent of their ranges; for it will even de- 
termine the resistance of the air, by causing the ball to strike 
the block of wood at different distances from the gun, thus 
showing the velocity lost by passing through those different 
spaces of air; all which circumstances are partly shown in 
my 4to. vol. of Tracts published in 1786, and which will be 
completed in my new volumes of miscellaneous tracts now 
printing. | / 

Lastly, in prob. 17, Prac. Ex. on Forces, some results of the 
same kind of experiment are successfully applied to deter- 
mine the curious circumstances of the first force or elasticity 
of the air resulting from fired gunpowder, and the velocity 
with which it expands itself. These are circumstances which 
have never before been determined with any precision. Mr. 
Robins, and other authors, it may be said, have only guessed at, 
rather than determined them. ‘That ingenious philosopher, by 
asimple experiment, traly showed that by the firing of a par- 
cel of gunpowder, a quantity of elastic air was disengaged, 
which when confined in the space only occupied by the pow- 
der before it was fired, was found to be near 256 times stronger 
than the weight or elasticity of the common atmospheric air. 
He then heated the same parcel of air to the degree of red hot 
iron, and found it in that temperature to be about 4 times as 
strong as before ; whence he inferred, that the first strength of 
‘he inflamed fluid, must be nearly 1000 times the Sls 


ore 


446 THEORY AND PRACTICE 


the atmosphere. But this was merely guessing at the degree 
of heat in the inflamed fluid, and consequently of its first 
strength, both which in fact are found to be much greater. — It 
is true that this assumed degree of strength accorded pretty 
well with that author’s experiments ; but this'seeming agree- 
ment, it may easily be shown, could only be owing to the in- 
accuracy of his own further experiments; and, in fact, with 
far better opportunities than fell to the lot of Mr. Robins, we 
have shown that inflamed gunpowder is about double the 
strength that he has assigned to it, and that it expands itself 
with the velocity of about 5000 feet per second. 

Fully sensible of the importance ef experiments of this 
kind, first practised by Mr. Robins with musket.balls only, my 
endeavours for many years were directed to the prosecution 
of the same on a larger scale, with cannon balls ; and [ hav- 
ing had the honour to be called on to give my assistance at 
several courses of such experiments, carried on at Woolwich 
by the ingenious officers of the Royal Artillery there, under 
the auspices of the Masters General of the Ordnance, I have 
assiduously attended them for many years. The first of these 
courses was performed in the year 1775, being 2 years after 
my establishment in the Royal Academy at that place : and in 
the Philos. Trans. for the year 1778, I gave an account of these 
experiments, with deductions, in amemoir, which was honour- 
ed with the Royal Society’s gold medal of that year. In con- 
clusion, from the whole, the following important deductions 
were fairly drawn and stated, viz. , 

Ist, It is made evident by these experiments, that gun-: 
powder fires almost instantaneously. 2dly, The velocities 
communicated to shot of the same weight, with different 
charges of powder, are nearly as the square roots of those 
charges. 3dly, And when shot of different weights are fired 
with the same charge of powder, the velocities communicated 
to them, are nearly inthe inverse ratio of the square roots 
of their weights. . 4thly, So that, in general, shot which are 
of different weights, and impelled by the firing of different 
charges of powder, acquire velocities which are directly as 
the square roots of the charges of powder, and inversely as 
the square roots of the weights of the shot. 5thly, It would 
therefore be a great improvement in artillery, occasionally to 
make use of shot of a long shape, or of heavier matter, as 
lead; for thus the momentum of a shot, when discharged 
with the same charge of powder, would be increased in the 
ratio of the square root of the weight of the shot; which. 
would both augment proportionally the force of the blow with. 

. which 


OF GUNNERY. 447 


which it would strike, and) the extent of the range to which 
it would go. 6thly, it would also be an improvement, to di- 
minish the windage ; since by this means, one third or more 
of the quantity of powder might be saved. 7thly, When the 
improvements mentioned in the last two articles are consider- 
ed as both taking place, it appears that about half the quantity 
ef powder might be saved. But, important as the saving may 
be, it appears to be still exceeded by that of the guns: for 
thus a small gun may be made to have the effect and execution 
of another of two or three times its size in the present way, 
by discharging a long shot of 2 or 3 times the weight of its 
usual ball, or round shot ; and thus a small ship might employ 
shot as heavy as those of the largest now in use. | 

Finally, as these experiments prove the regulations with re- 
spect to the weight of powder and shot, when discharged from 
the same piece of ordnance ; so, by making similar experi- 
ments with a gun varied in its length by cutting off from it a 
certain part, before each set of trials, the effects and general 
rules for the different lengths of guns, may be with certainty 
determined by them, Inshort the principles on which these 
experiments were made, are so fruitful in consequences, that, 
in conjunction with the effects of the resistance of the medi- 
um, they appear to be sufficient for answering all the inqui- 
ries of the speculative philosopher, as well as those of the 
practical artillerist. was} | 

Such then was the summary conclusion from the first set of 
“experiments with cannon balls, in the year 1775, and such 
were the probable advantages to be derived from them. Iam 
not aware however that any alterations were adopted from 
them by authority in the public service: unless we are to 
except the instance of carronades, aspecies of ordnance that 
was afterwards invented, and in some degree adopted in the 
public service ; for, in this instance, the proprietors of those 
pieces by availing themselves of the circumstances of large 
balls, and very small windage, have, with small charges of 
powder, and at little expense, been enabled to produce very 
considerable and useful effects with those light pieces. 

The 2d set of these experiments extended through most 
part of the summer seasons of the years 1783, 1784, 1785, 
and some in 1786. The objects of this course were nume- 
rous and various: but the principal articles as follow: 1. The 
velocities, with which balls are projected by equal charges of 
powder, from pieces of equal weight and calibre, but of dif- 
ferent lengths. 2. The velocities with different charges of 
powder, the weight and length of the guns being equal. 3. 
The greatest velocities due to the different lengths of guns, 


to 


448 THEORY AND PRACTICE 


‘to be, ascertained by successively increasing the charge, till 
the bere should: be filled, or till the velocity should decrease 
again. 4. The effect of varying the weight of the piece ; 
every thing else being the same. 5. The penetrations of 
balls into blocks of wood. 6. The ranges and times of flight 
of balls ; to compare them with their first velocities, for as- 
certaining the resistance of the medium. 7. The effect of 
wads; of different degrees of ramming, or compressing the 
charge ; of different degrees of windage ; of different positions 
of the vent; of chambers: and trunnions, and every other 
circumstance necessary to be known for the improvement of 
artillery. 

An ample account is given of these experiments, and the 
results deduced from them in my volume of Tracts published 
in 1786 ; some few circumstances only of which can be noted 
here. In this course, 4 brass guns were employed, very 
nicely bored and cast on purpose, of different lengths, but 
equal in all other respects, viz. in weight and Hore, &e. The 
lengths of the bores of the guns were, ~ 

the gun n° 1, was 15 calibres, length of bore 28-5 inc. 

fb ODF 2 AS OD Cah eres, ses JG ed GO 
sora n? Sy oS B0°calibres, (42S 9S BR 

ws hay? Qe oS) AO eah brags a5 054 Se gOS 
the calibre of each being 23, inches, and the medium weight 
of the balls 16 oz. 13 drams.. 

The mediums of all the experimented velocities of the 
balls with which they struck the pendulous block of wood, 
placed at the distance of 32 feet from the muzzle of the gun, 
for several charges of powder, were as in the following table, 


Tabie of Initial Velocities. 


Powder. The Guns. 




















oz. |No. 1. |No. 2. |No. 3. |No. 4. 
2 780 | 835 | 920| 9701 
4 1100 | 1180 | 1300 | 13°70 } 
6 1340 | 1445 | 1590 | 1680 
8 1430 | 1680 } 1790 | 1940 
12 1436 | 1640 5 : 


14 ; 1660] . : 
16 : goer AeBOOD 
eos Reon, Son? : 2200 


placed in the ist column, for all the four guns, the num- 
bers denoting so many feet per second.» Whence in general 
if 


OF GUNNERY. 449 


it appears how the velocities increase with the charges of 
powder, for each gun, and also how they increase as the guns 
are longer, with the same charge, in every instance. 

By increasing the quantity of the charges continually, for 
each gun, it was found that the velocities continued to increase 
till they arrived at a certain degree, different in each gun ; 
after which, they constantly decreased again, till the bore was 
quite filled with the charge. The charges of powder when 
the velocities arrived at their maximum or greatest state, were 
various, a8 might be expected, according to the lengths of the 
guns ; and the weight of powder, with the length it extended 
in the bore, and the fractional part of the bore it occupied, 
are shown in the following table, of the charges for the great- 
est effect. - 


| | The Charge. 


ee SS 








Length |Weight,| Length. 
of the —_——— 


Gun, 








laches, Part of 

















n°. ; Bore OZ. 

whole 
1 | 98-5| 12 | 82 | * 
2) 38-4| 14 95 | 3 
s 57-71-16 10°7 | 7 
4] 80-21 18 | 121 | 3, 








Some few experiments in this course were made to obtain 
the ranges and times of flight, the mediums of which are ex- 
hibited in the following table. 






































Pow- Balls. Elevat. | Time First 
Guns er of fight.| 298° | vel 

~ |Weight.| Diam. as reas bd stand 

oz. | oz. dr. | inch. secs. feet. | feet. 

n°2 2 | 16 10 1°96 45° 21-2 | 5109 863 

do. S16. 6 1:96 15 9°2 | 4130 868 

dP 4 | 416 8 1:96] 15 | 9-2 | 4660 | 1234 

do. 8 16 12 1°96 15. 14-4 | 6066 | 1644 

do. 12-7" 162 26 15 15°5 | 6700 | 1676 

n°3.} 8 | 15 ~8| 1:96] 15 | 10:1 | 5610 | 1938 

















In this table are contained the following concomitant data, 
determined with a tolerable degree of precision; viz. the 
weight of the powder, the weight and diameter of the ball, 
the initial or projectile velocity, the angle of elevation of the 

Vou. II, 58 | gun, 


450 THEORY: AND PRACTICE 


gun, the time in seconds of the ball’s flight through the air, 
and its range, or the distance where it fell on the horizontal 
plane. From which it is hoped that some aid may be derived 
towards ascertaining the resistance of the medium, and its 
effects on other elevations, &c. and so afford some means of » 
obtaining easy rules for the cases of practical gunnery. 
Though “the completion of this enquiry, for want of time at 
present, must be referred to another work, where we may 
’ have an opportunity of describing another more extended 
course of experiments on this subject, which have never yet 
been given to the public. 
_ Another subject of enquiry’ [Penetrations of Balls into 
in the foregoing experiments, solid Elan wood! 


was, how far the bails would Pander ’| a bor. 





penetrate into solid blocks of 
elm wood, fired in the direc- 
tion of the fibres. ‘The an- 
nexed tablet shows the results 
of a few of the trials that | 








were made with the gun n° 2, 
with the most frequent charges 
of 2, 4, and 8 ounces of pow- 
der ; and the mediums of the 
penetrations, as placed in the 
last line, are found to be 7, 15, and 20 cable with those 
charges. These penetrations are nearly as, the numbers 

2, 4,6, or 1, 2,3; but the charges of powder are as 

2, 4, 8, or 1, 2, 4; so that the penetrations are propor- 
tional to the charges as far as to 4 ounces, but in a less ratio 
at 8 ounces; whereas, by the theory of penetrations the 
depths ought to be proportional to the charges, or which is 
the same thing, as the squares of the velocities. So that it 
seems the resisting force of the wood is not uniformly or con- 
stantly the same but that it increases a little with the increas- 
ed velocity of the ball. This may probably be occasioned by 
the greater quantity of fibres driven before the ball; which 
_may thus increase the spring and resistance of the wood, and 
prevent the ball from penetrating so deep as it otherwise 
might do. 

From a general inspection of this second course of these 
‘experiments, it appears that all the deductions and observa- 
tions made on the former course, are here corroborated and | 
strengthened, respecting the velocities and weights of the 
balls, and charges of powder, &c. It further appears also 
that the velocity of the ball increases with the increase of 


charge 














OF GUNNERY. 451 


charge only to a certain point, which is peculiar to each gun, 
where it is greatest; and that by further increasing the 
charge, the velocity gradually diminishes, till the bore is 
quite full of powder. That this charge for the greatest ve- 
locity is greater as the gun is longer, but yet not greater in 
so high a proportion as the length of the gun is; so that the 
part of the bore filled with powder, bears a less proportion to 
the whole bore in the long guns, than it does in the shorter 
Ones: the part which is filled being indeed nearly in the 
inverse ratio of the square root of the empty part. __ 

It appears that the velocity, with equal charges, always in- 
creases as the gun is longer; though the increase in velocity 
is but very small in comparison to the increase in length; the 
velocities being in a ratio somewhatess than that of the square 
roots of the length of the bore, but greater than thai of the 
cube roots of the same, and is indeed nearly in the middle ra- 
tio between the two. 7 | : 

It appears from the table of ranges, that the range increas- 
es in a much lower ratio than the velocity, the gun and ele- 
vation being the same. And when this is compared with the 
proportion of the velocity and length of gun in the last para- 
graph, it is evident that we gain extremely little in the range 
by a great increase in the length of the gun with the same 
charge of powder. In fact the range is nearly as the 5th root 
of the length of the bore: which is so small an increase, as 
to amount only to about a 7th part more range for a double 
length of gun.—F rom the same table it also appears, that the 
time of the ball’s flight is. nearly as the range; the gun and 
elevation being the same. 

It has been found, by these experiments, that no difference 
is caused in the velocity, or range, by varying the weight of 
the gun, nor by the use of wads, nor by different degrees of 
ramming, nor by firing the charge of powder in different 
parts of it. But that a very great difference in the velocity 
arises from a small degree in the windage : indeed with the 
usual established windage only, viz. about ,, of the calibre, 
no tess than between 1 and } of the powder escapes and is 
lost : and as the balls are often smaller than the regulated size, 
it frequently happens that half the powder is lost by unneces- 
sary windage. | 

It appears too that the resisting force of wood, to balls fired 
into it, is not constant : and that the depths penetrated by balls, 
with different velocities or charges, are nearly as the logarithms 
of the charges, instead of being as the charges themselves, or, 
which is the same thing, as the square of the velecity.—Last- 
ly, these and most other experiments, show, that balls are 

. greatly 


462 THEORY AND PRACTICE 


greatly deflected from the direction in which they are pro- . 


jected ; and that as much as 300 or 400 yards in a range of a 
mile, or almost 1th of the range. 
We have before adverted to a third set of experiments, of 


still more importance, with respect to the resistance of the | 


medium, than any of the former ; but, till the publication of 
those experiments we cannot aval parkelyes of all the disco- 
veries they contain. In the mean time however we may ex- 
tract from them the three following tables of. resistances, for 


three different sizes of bails, and for velocities between 100. 


feet and 2000 feet per second of time. 


TaBLE [ Tape Il. TABLE qt. 
Resistances to a ball of 1-965)| Resist. to a ball||Resist. to a ball 
| inches diameter, and 16 0z.||2°78 india and}|3-55 m. dia. and 


13 dr. weight. eae 3 1b. weight. _ \\6lb. loz. 8dr. wt. 
Vel. | Resistances. {1 Dif. 12d Dif|} Vel. | Res. [Difs.|| Vel. | Res. | Difs. 
feet |. lbs. ozs. feet } Ibs. feet | Ibs. 

100} 0°14 23). 81) 52 900} 35 | 6 }}1200}115 9 
200} 0°69} 11 | 14 6 950} 41 | 6 4)1250)124 9. 
300} 1°56} 25 | 20 7 {}1000} 47 | 6 }/1300)133 | 9 
400} 2°81] 45 |.27 8 ||}1050| 53 | 7 4111350);142 | 10 
500} 4°50; 72 | 35 9 111100} 60 | 7 1}1400/152 | 10 
600} 6°69} 107 | 44°] 10 |}1150) 67 | 7 |)1450)162.4 102 
700}. 9°44| 151 | 54 | 12 11/1200) 74] 8 |]1500 1721) 11d 
860/12°81) 205 | 66 | 13 {{1250) 82 | 9 1115501184 | 13 
900/16°94} 271 | 79 | 13. 111300) 91 {10 }}1600}197 | 14 
1000/21°88) 350 | 92°] 12 i1350}101 |11 j|1650/211 | 15 


1100/27°63} 442 |104 | 11 |}1400)112 |1021|1700/226 | 16 
1200/34°13} 546 |115 | 9 |11450|1222110 1/1750/242 | 17 
1300/41°31} 661 1124 | 7 f11500)1324) 9 1/1800/259 
1400149°06| 785 1131 | 4 1/1550)1414) 82 | 
1500|57°25| 916 |135 | 0 |11600/150 | 8 
1600/65:69}1051 |135 |—2 |11650|158 | 7 
1700|74:13)1186 |133 |—5 |11700/165 | 6 
1800/82°44/1319 |128 |—6 }/1750/171 | 5 
1900/90:44/1447 |122 | 1800|176 

2000/98-0611569 | 





PROBLEM If. 


To deterrune the Resistance of the Medium against a Ball of 
any other size, moving with any of the Velocities given in the 
foregoing Tables. 


The analogy among the numbers in all these tables is 
very remarkable and uniform, the same general laws running 
threape 


OF: GUNNERY. 453. 


through them all.. .The same laws are also observable as in 
the table of resistances in page 412 of this volume, particu- 
larly the ist and 2d remarks immediately following that table, 
viz. that the resistances increase in a higher proportion than 
the square of the velocities, with the same body ; and that 
the resistances also increase in a rather higher ratio than the 
surfaces, with different bodies, but the same velocity. Yet 
this latter case, viz. the ratios of the resistances and of the 
surfaces, or of the squares of the diameters which is the same 
thing, are so nearly alike, that they may be considered as equal 
to each other in any calculations relating to artillery practice. 
For-example, suppose it were required to determine what 
would be the resistance of the air against a 24]b ball dechinged 
with a velocity of 2000 feet per second of time. Now, by 
the first of the foregoing tables, the ball of 1-965 inches dia- 
meter, when moving with the velocity 2000, suffered a resist- 
ance of 98Ib: then. since the resistances, with the same velo- 
city, are as the surfaces ; and the surfaces are as the squares 
of the diameters ; and the diameters being 1:965 and 5:6, the 
ms ee of which are 3°86 and 31-36, therefore as 3°86 : 31:36 

: 98lb : 796]1b; thatis, the 24lb ball would suffer the enor- 
mous resistance of 796lb in its flight, in opposition to the di- 
rection of its motion! 

And, in general. if the diameter of any. eee ball be 
denoted by d, and r denote the resistance in the Ist table due 


; d2 
to the proposed velocity of the 1°965 ball; then sas will de- 


note the resistance with the same velocity against the ball 
whose diameter is d; or it is nearly 1d?r which is but the 
28th part greater than the former. 


PROBLEM II. 


To assign a Rule for determining the Resistance due to any es 
* determinate Velocity of a Given Ball. 


This problem is very difficult to perform near the rath on 
account of the variable ratio which the resistance bears to the 
velocity, increasing always more and more above that of the 
square of the velocity, at least toa certain extent; and indeed 
it appears that there is no single integral power whatever of 
the velocity, or no expression of the velocity in one term only, 
that can be proportional to the resistances throughout. — It is 
true indeed that such an expression can be assigned by means 

of a fractional power of the velocity, or ee one whose 
index is a mixed number, viz, 2;'5 or 2:1; thus —— vices = the 
resistance, 


454. THEORY AND PRACTICE 


resistance, is a formula in one term only, which will answer 
to all the numbers in the first table of resistances very nearly, 
and consequently, by means of the ratio of the squares of the 
diameters of the balls, for any other balls whatever. This 
formula then, though serving quite well for some particular 
resistance, or even for constructing a complete series or table 
of resistances, is not proper for the use of problems in which 
fluxions and fluents are concerned, on account of the mixed 
number 2,1., in the index of the velocity V. 

We must therefore have recourse to an expression in two 
terms, or a formula containing two integral powers of the ve- 
locity, as v* and v, the first and 2d powers, affected with ge- 
neral coefficients m and 2, as mv? +- nv = r the irbuistabee: 
Now, to determine the general numerical values of the coef- 
ficients m and n, we must adapt this general expression mv? 
-+- nv = r, te two particular cases of velocity at a convenient 
distance from each other, in one of the foregoing tables of re- 
sistances, as the first for instance. Now, after making seve- 
ral trials in this way, I have found that the two velocities of 
500 and 1000 answer the general purpose better than any 
other that has been tried. Thus then, employing these two 
cases, we must first make » = 500, and r = 41\Ib, its corres- 
pondent resistance, and then again v= 1000, and r=21: 88lb, 
the resistance belonging to it; this will give two equations, 
by which the general value of mand of nwill be determined. 
Thus then the two equations being 

5002m-}+ 600n=4:-5, 
and 10002m-+ 1000n=21:88 ; 
dividing the Ist by 500, and the § 500m-+-n=-009, 

2d by 1000, they are 1000m-+-n=:02188 ; 
the dif. of these is. . . .  .. 500m=-01288, 
and therefore div. by 500, gives m =+00002576 ; 
hence n = :009 — 500m = -009 -— -01288 = — -00388 = n. 
Hence then the general formula will be :00002576v? — 
‘00388v=r the resistance nearly in avoirdupois pounds, in all 
cases or all velocities whatever. 


_ Now, 


OF GUNNERY. “ABB 


























Now, to find how near to the \Velors.|Comput.| Exper. | 
truth this theorem comes, in | ory, | resists | resists 
every instance in the table, by © 7h oe 
substituting for v, in this formula, 100} 13’: | “17 | 
all the several velocities. 100, 200) —--25 +} +69] 
200, 300, &c. te 2000, these give 3001 4:15 | tre 
the correspondent values of r, or 400] 2°57 9-8] 
the resistances, as in the 2d co- a 4°50 4:50 
lumn of the annexed table, their | = gQo| 6-94 6°69 
velocities being in the first co- 8 9:90 |. aia | 
lumn ; and the real experimented | ~~ 9090) 13-38 |! 19-81) 
resistances are set opposite to 320 17°37 | 16°94 
them in the 3d or last column of 10e0| 21-88 | 21-88 
the same. By the comparison of He 26-90 | 27-63 
the numbers in these two co- 1900! 32°44 | 34:13 
lumns together, it is seen that sere 39:49 | 41.3] 
there are.no where any great dif- 1400] 45-06 | 49-06 
ference between them, being 1500! 52-14 | ee 
sometimes a little in excess, and 1600| 59:74. | 65-69 
again a little in defect, by very 1700] 67°85 | 14-13 
small differences ; so that, on the 1800] 76-48 | 82:44 


a ball, in all degrees of its elocit from the first or greatest, 
to the smallest er last. Except in the first two or three num- 
bers, at the beginning of the table, for the velocities, 100, 
200, 300, for which cases another theorem may be employed. 
Now, in these three velocities, as well as in all that are 
smaller, down to nothing, the theorem *00001725v2 = r the 
resistance, will very well serve, as it brings out for the first 
three resistances -17 and 69 and 1°55}, differing in the last 
only by a very small fraction. ! 


Corol. 1. The foregoing rule -000025760? — -003880 = r, 
denotes the res sistance: for the ball in the first table, whose 
diameter is 1:965, the square of which is 3°86 or almost 4 ; 
hence to adapt it to a bali of any other diameter d, we have 
only to alter the former in proportion to the squares of the 


d as dd : 
diameters, by which it becomes 3 G6 \ 000025760? — °00388v) 


= (9000066702 — ‘OU1v)d? = (0000020? — ‘001v)d2, which 
is the resistance for the ball whose diameter is d, with the 
velocity v. | 


Corol. 2. And in a similar manner. to adapt the theorem 
‘§0001725u? = r, for the smaller velocities, to any other size 
of 


e 


456 | THEORY AND PRACTICE 


of ball, we rnisi multiply it Dy aor a , the ratio of the surfaces, 


by which it becomes ona ees # =r. 
We shall soon take occasion to make some applications i in 
the use of the foregoing formulas, after Conslce eine the eflects 
-of such velocities in the cases of honresistances. 


PROBLEM IIL. 


Lo determine the Height to which.a Ball will rise, when jired 
from a cannon Perpendicularly Upwards with a Given Velo- 
city, m a Nonresisting Medium, or supposing no Resistance 
in the Avr. 


By art. 73, Motion and Forces, this vol. it appears that any 
body projected upwards, with a given velocity, will ascend to 
the height.due to the velocity, or the height from which it 
must naturally fall to acquire that velocity ; and the spaces 
fallen being as the square of the velocities ; also 16 feet being 
the space due to the velocity 32; therefore the space due te 
any proposed velocity v, will be Age thus, as 322 : 16:1 v2 zs 
the space or as 64: 1 :: v? : J; v2? = s the space, or the 
height to which the velocity v wilt cause the body to rise in- 
dependent of the air’s resistance. 

Exam. For example, if the first or projectile velocity; be 
- 2000 feet per second, Deine nearly the aresier experimented 
velocity then the rule ;! v2 == s becomes zy X 20007 = 62500 
feet == 112 miles: that is, any body. projected with the ve- 
locity 2000 feet, would ascend nearly 12 miles in height, with- 
out resistance. 

Corol. Because, by art. 88 Projectiles this vol. the greatest 
range is just double the height due to the projectile velocity, 
therefore the range at an elevation of 45°, with the velocity 
in the last example, would be 232 miles in a nonresisting me- 
dium: We shall now see what the effects will be with the 

resistance of the air. 


“PROBLEM IV. 


To determine the Height to which a Ball projected Upwards, as 
an the last problem, will ascend, being Resisted 3° the Atmos- 
phere. 


Putting x to denote any vakiable and increasing height as- 
cended by the ball; v its variable and decreasing velocity there; . 
d the diameter of the ball, its weight being w ; m= -000002, 
and n = ‘001, the co- -efficients of the two terms denoting the 
law of the air’s resistance. Then (mv? ~nv)d?, by cor, 1 to 

prob. 


OF GUNNERY. — A5% 


prob. 2, will be the resistance of the air against the ball in 
avoirdupois pounds: te which if the weight of the ball be 
added, then (mv? — nv) d? -+-w will be the whole resistance to 
the ball’s motion ; this divided by », the weight of the ball in 


. 25 2 am 
(mo? = lat +e _ me - rhe Boies Pihe ro 


motion, gives ~—— 
tarding force. Hones the general formula vy =2g fx (theor. 
18 ore see 
10 pa. 379 this volume) becomes ~vp=2g2x-— un —— me 
making v negative because v is s decreasing, where g =16 ft. ; 
and Se . Ear 
vv Se tke vy 





pee 3 * Gino? —nv)d?: rear 2egmd? i n w 
pt — —v + -— 
m md? 


Now, for the easier finding the fluent of this, ssa D ox 


n : 
pa, then v= 24+ 7, and 0? = ae Fag and 20 
m 

et , 


(k0 
meee eg, and vt of LA and'e? a em fe 
2 am m 4m? m 








yore these being substituted in the above value of z, it be- 
comes 7 = ; 
—x% ‘ 22+ 972 nae: om WY ge) cee ; —w ¥ 
Semda na. wy Pend - ~ Semi 
ee Pei gee ape Ml cst cert Rem 
zz -+pz 





w . as 
WEG = putting p=, and q? ra ase 2, or p? +7.) 


Then the general fluents, taken aby the 8th and 11th forms 
of the table of Fluents give “tana x [Zlog. (22 ae i 


X arc to rad. g and tan. 2] = ———- Se =a X [dlog. (wv? othe = Feat 
2s. 9a X arc to rad. g and tang. v—p]. But, at the Nahe 


of the motion, when the first velocity i is v for instance, and the 
space zis = 0, this fluent becomes 


ster SN eB P 
X[plog. So fcteetomi + )+4) Xare radius q tan. 


v= p.] Hence by subtraction, and taking » = 0 Ofer the end 


of the motion, the correct fluent becomes «=. X[Jlog 
Von dbs Xa) ies SHO. esis (yaad 


458 THEORY AND PRACTICE 


Se Soe ee Ce P 
aha 8 SADR las = fo 
(v3 hs ree log. — Fue X (arc tan. v—p— arc tan. 
—p torad. q)]. 
But as part of this fluent, denoted by.=- P yx the dif. of the 


~ two arcs to tans. v — p and —p, is always very small in com- 
parison with the other preceding terms, they may be emitted 
without material error in any ier esi instance ; and then the 
w 
v? eo = vp 


. md? , for the ut- 











w 
fluent is a= Timalt hyp. log. 


md? 
most height to which the ball will ascend, when its motion 
ceases, and i is stopped, partly by its own gravity, but chiefly 
by the resistance of the air. 
But now, for the nme value of the general coefficient 


W 


pas S and the term —— ;: because the mass of the ball to the 


ai 
sakes d, is *5236sd3, if its mee gravity be s its weight 


will be -5236sd3 = w ; therefore — af = °5236sd, and —— a —= 


78540sd, this divided by 4g or 64 it gives ——. = 1297-2sd 
gm 

for the value of the general coefficient, to any diameter d and 

specific gravity s. And if we further suppose the ball to be 

cast iron, the specific gravity, or weight of one cubic inch of 

which is °26855, it becomes 330d, for that coefficient ; also 


78540sd=21090d = Liat, , and —=150. And hence the fore- 


md? a : 
going fluent becomes 330d x nett log. bf Ld 2a 


v2 —150v-+21090d 


~ "910900. 
bolic for the common logs. And this is a general expression 
for the altitude in feet, ascended by any iron ball, whose dia- 
meter is d'inches, discharged with any velocity v feet. So 
that, substituting any values of d and v, the particular heights: 
will ‘be given to which the balls will ascend. which it is evi- 
dent will be nearly in proportion to the diameter d. 


Exam. 1. Suppose the ball be. that belonging to the first 
table of resistances, its weight being 16 oz. 13 dr. or 1-05 Ib, 
and its diameter 1-965 inches, when discharged with the velo: 
city 2000 feet, being nearly the greatest charge for any iron 
ball. The calculation being made with these values of d and 

v, the 





or 760d X com. log, —— changing the. hyper- 


OF GUNNERY. 459 


v, the height ascended is found to be 2920 feet, or little more 
than half a mile; thoagh found to be almost 12 miles with- 
out the air’s resistance. And thus the height may be found 
for any other diameter and velocity. 

Exam. 2. Again, for the 24lb ball, with the same velo- 
city 2000, its diameter being 5-6 =d. Here 760d = 4256 ; 

v2?— 150v-+21090d 38181 ‘igthe aes 

and sd a a Serer the log. of whichis 1°50958 ; 
theref. 1:50858 X 4256 = 6424 = x the height, being a lit- 
tle more than a mile. 

We may now examine what will be the height ascended, 
considering the resistance always as the square of the velocity. 


PROBLEM V. 


To determine the Height ascended by a Ball projected as in the 
two foregoing problems ; supposing the Resistance of the Air 
to be as the Square of the Velocity. 


Here it will be proper to commence with selecting some 
experimented resistance corresponding to a medium kind. of 
velocity between the first or greatest velocity and nothing, 
from which to compute the other general resistances, by con- 
sidering them as the squares of the velocities. It is proper 
to assume a near medium velocity and its resistance, because, 
if we assume or commence with the greatest, or the velocity 
of projection, and compute from it downwards, the resistances 
will be every where too great, and the altitude ascended much 
less than just ; and, on the other hand, if we assume or com- 
mence with a small resistance, and compute from it all the 
others upwards, they will be much too little, and the com- 
puted altitude far too great. But, commencing with a me- 
dium degree, as for instance that which has a resistance 
about the half of the first or greatest resistance, or rather a 
little more, and computing from that, then all those computed 
resistances above that, will be rather too little, but all those 
below it too great ; by which it will happen, that the defect of 
the one side will be compensated by the excess on the other, 
and the final conclusion must be near the truth. 

Thus then, if we wish to determine, in this way, the alti- 
tude ascended by the ball employed in the Ist table of resist- 
ances when projected with 2000 feet velocity ; we perceive 
by the table, that to the velocity 2000 corresponds the re- 
sistance 98Ib; the half of this is 49 to which resistance 
corresponds the velocity 1400, in the table, and the next 
greater velocity 1500, with its resistance 571, which will be 
properest to be employed here. Hence then, for any other 

. ; velocity 


7 


460- THEORY AND PRACTICE 


: velocity v, in general, it willbe, according to the law of the 
67103. 

1500? 
00002540? = av?, putting o = +0000254, which will denote 
the air’s resistance for any velocity v, very nearly, animate 
“from 2000. 

Now let x denote the altitude ascended whe the velocity, 
_is v, and w the weight of the ball: then, as above, av?, 1s the 
resistance deo the alr, hence av? -}- w is the whole resisting 


squares of the velocities, as 15002 bya tc eerie 





w 
force, and = aE a = jf the retarding force; 
therefore — vv = oefeme X 24%; 
and hence x = arab Oe TROY se 


ane 


2 av? -+w ra 
§ ss ate 


the fluent of which, by form 8, is a2 x typ. log. (08 = ys 


which when «= o and v = v the fit or projectile veluctty, 





=); theref. by Aaa 


, ave--w 
the correct no is ¢ = = he. pare the height x 
when the velocity is reduced to » ; asd aes » = 0, or the 





velocity is quite exhausted this becomes _- hele a8 
for the whole height to which the ball will ascend. 

Ez. 1. The values of the letters being w= 1°05], 4g=64, 
a= "0000255, the last expression becomes 645 X hyp. log. 
v? 41266 © wi ara 1266: > 
Rats - And th 

41266 41266 here:the 
first velocity v being 2000, the same expression 1484 xX log. 


2+-41266 
: ion ial becomes 1484 D4 log. of. 97: 93 = 2955 for the 


height ascended, on this hypothesis ; which was 2920 by the 
former problem, being nearly the same. 

Ex. 2. Supposing the same ball to be projected with the 
velocity of only 1500 feet. Then taking 1100 velocity, whose 
tabular resistance is 27:6, being next appre the half of that 
for 1500. Hence, as 1100? : v2 :: 276 : -00002375v2 =av?. 
av? +w 


—- 


——, or 1484 x com. log. - 





This value of a substituted i in 1 the b corods 5 he 1., 
SO 

aléy 1500 for ¥, and. 1: (05 for w, it brings out 2 == 2728 for 
the 


a 
j OF GUNNERY. 463 


the height in this case, being but a little above the ratio of 
the square roots of the velocities 2000 and 1500, as that ratio 
would give only 2560. 

Ex. 3. To find the height ascended by the first ball pro- 
jected with 860 feet velocity. Here taking #00, prboke * 
sistance 6°69 is a near medium ; then as 600? : 6°69: 


-0000186 = a. Hence ax page ae ne 


which is less than half the range (5100) at 45° elevation, but 
more than half the range (4100) at 15° elevation, art. 105 of 
Motion and Forces, being indeed nearly a medium between 
the two. 

Ex. 4. With the same ball, and 1640 velocity. Assume 
1200, whose resistance 34:13 is nearly a medium. Then as 


2 3 
1200? : 34:13 :: 1 : :0000237 =a. Hence = -xh.l.- sesh 
W 


== 2854; again less than half the range (6000) by experiment 
in this vol. even with 15° elevation. 
Ex. 5. For any other ball whose diameter is d, and its 

















: ad?y2 22 
Ween the resistance of the air being Sage Tana == bd? 
v patting b= 7 ae the retarding force will be at eg 
thence eres a a at, and 7== ae). a and 
the cor. flu. c= eee xh. Leora at ene Xh. 1. — 


for the whole height when v=0. Now if the ball be a 24 
pounder, whose diameter i is 5: Bs and its square 31-36 ; then 


62°72 24 3 
j2 = ~==-0002091, and ——_— ee ee 
bd = 300000 Ds ars Tpabat ober 
bd?2v2-+-w °836-+24 _ 860. 215 
and bd?-v2 =836, and fe ee Tea! a on ==" there: 
w 94° ~*~ Q4 6 


ran 
Pre, 17904 SK, salle SS Tey 357888 == 6420 ; being 








more than double the hott of that of the small ball, or a lit- 
tle more than a mile, and very nearly the same as in the 2d 
ree to peel. 4, 


PROBLEM VI. 


To determine the Time of the Ball’s ascending to the Height de- 
termined in the ss pee, Py oe same Projectile Velocity as - 


theve given, PV acta ee 


7 


yo 


462 THEORY AND PRACTICE 





side aaa —%wW ” 
By that prob. xc -X —, ther. oe crt —— ; 
250 ae ae ore 
a 


the fluent of which, by form 11, is = JS -- X arc to ra- 


: ? ~ 1 
dius } tang. Ean en ed = x arc tan. 


ve ve 


a 


; or by correc- 








— arc tang. —), the time 
Vv i Vs 


in general when the first salucits vis reduced to v. And 


tion t= av 5 X (are tang. 


a : ; 1 
when v=0, or the velocity ceases, this becomes t= ava. 





arc to tang. for the time ef the whole ascent. _ 


J 
Now, as in the last Lav y==2000, pai 05, a = -000025¢ 
229 


Rao EaT —e _- = —_~_ = 9 
9000000 Hence — - 41266, and «/ — . 03: 14, and 


os = 98-445 the tangent, to which corresponds the arc 
J = ee 

> 1 
of 89° 25’, whose length is 1°5606 ; pe X 203:14 X 


3°14 ~ 1°-5606 : 
1°5606 = ee = 9-91, the whole time of ascent. 


Remark. The time of freely ascending to the same height 
2955 feet, that is, without the air’s resistance, would be 


Jan = 1 ,/ 2955 = 1359; and the time of freely as- 


cending, commencing with the same velocity 2000, would be. 
v 2000 
4 1’ op 
bg (320 ae if 
. . PROBLEM VII. 
To determine the same as in prob. v, taking into the uccount 


the Decrease of Density in the Air as the Ball ascends an the 
Atmosphere. 


In the preceding problem relating to the heipht a and time 
of balls ascending in the atmosphere, the decrease of density 
in 


OF GUNNERY. 463 


in the upper parts of it has been neglected, the whole height 
ascending by the ball being supposed in air of the same den- 
sity as at the earth’s surface. But it is well known that the 
atmosphere must and does decrease in density upwards, in a 
very rapid degree ; so much so indeed, as to decrease in geo- 
metrical progression : at altitudes which rise only in arithme- 
tical progression : by which it happens, that the altitudes as- 
cended are proportional only to the logarithms of the decrease 
of density there. Hence it results, that the balls must be al- 
ways less and less resisted in their ascent, with the same velo- 
city, and that they must consequently rise to greater heights 
before they stop. It is now therefore to be considered what 
may be the difference resulting from this circumstance. 

Now, the nature and measure of this decreasing density, 
ef ascents in the atmosphere, has been explained and deter- 
mined in prop. 76, Pneumatics. It is there shown, that 
if p denote the air’s density at the earth’s surface, and 
@ its density at any altitude a@ or x, then is + = 63551 x 


log. of ; in feet, when the temperature of the air is 55°; 
and 60000 x log. ‘ for the temperature of freezing cold; 


we may therefore assume for the medium z= 62000 X log. 
D 


J for a mean degree between the two. 


But to get an expression for the density d, in terms of x 
out of logarithms, without which it could not be introduced 
into the measure of the ball’s resistance, in a manageable form 
we find in the first place, by a neat approximate expression 


for the natural number to the log. of a ratio a: whose terms 
do not greatly differ, invented by Dr. Halley, and explained 
cay 
in the Introduction to our Legarithms, p. 110, that Semi D 
2 
nearly, is the number answering to the log. / of the ratio-, 
where n denotes the modulus -43429448 &c. of the common 


logarithms. But, we before found that x = 62000 Xlog. of “i 


Bi; Bi 

oF 50008 the log. of 7 which log. was denoted by / in the 
expression just above, for the number whose log. is / or 
x 


62000 





ssoo8' substituting therefore for 1, in the expression 


n= il 


464 _ THEORY AND PRACTICE 








“ pap Aems: 
: 2 | BEOOS <=, Oa 
in X p, it gives the natural number —— XD=p or, 
pebisigee 
124000. 
124000n —« 
124000n-bx ==; the density of the air at the altitude «, put- 


ting p = 1 the density at mid surface. Now put 124000n or 





nearly 54000 =e; ; then © ~ will be the density of the air 


at any general height <. 

But, in the 5th prob. it appears that av? denotes the re- 
sistance to the velocity v, or at the height «, for the density 
of air the same as at the surface, nets is we ie in the 


ratio of c-+ x to c——«; therefore av* X mae will be the 
resistance at the height «, to. the velocity », where a == 
aes a To this. adding w, the weight of the ball, gives 


av? X ey eel m0) » for the whole resistance, sd from the air 


av? C2 
+= — will denote 


and the ball’s mass ; ; eonseq, —— X = a 
the Praise force of the ball. Or, if o include the 
~~, which will make 








small part —or 1, within the factor : 











cH+z 
no sensible difference 1m the result, but be a great deal simpler 
in the process, then is - ae x jae = f the accelerating 
2 
fie Conseq. aa ane = ae = 2gx2 X as li tie: and 
C—2x. woo =v i 2 
ence reat 2g t ae. .or by ‘division, — - 2 me pelea 

‘s W — vy Ee MS <; 4 
eats ? 

32 ; 

wl 02 wie 

oe 


Now ithe fluent of the first side of tne nas ae is evi- 
dently —« + 2c x h. 1. (c+-x); and the fluent of the latter 


side, the same as in prob. 5, is =X h. 1. (v8 +o; there- 
foye the general fluential equa. is — #-+ 2¢ x h. J, G + 2)= 


an x h. b (v ets —). But when «==0, and v=v the initial 


seat becomes 0+-2c x b. 1 (=o xh. 1. (v8 + =); 
there 


OF GUNNERY. 465 


theref. by subtraction the correct fluents are~-« + 2c X h. 1. 


: mse bi av? w ° ° - . 
Br tk , When the first velocity v is dimi- 








c. ©664a "" apitw 
nished to any less one v; and when it is quite extinct, the 
state of the fluents becomes — x + 2c xh. I. ene = = x 
. ! ¢ 
ay 
h. 1. se for the greatest height « ascended. 


gs 


Here, in the quantity h. 1, — . , the term x is always small 


in respect of the other term c ; Checetee: by the nature of 


logarithms, the h. A of Eig nearly = a ae 2 rs ee ; theref. 
Acx Zen? ei erase 


the above fl oe UOT BU 
ove fluents become — x + —— oop OES ewe © 
av? +w 


=—— xh.1,°%-™. Now the latter side of this equation 
64a w 

is the same value for x as was found in the Sth problem, which 
therefore put = es ee the value of x will be easily found _ 


from the formule: a z= b, by a quadratic equation. Or, 
still easier, and sufficiently near the truth, by agyantanng b 





for « in the numerator and the denominator Wad o or then 


pa ea = 6, and hence x = zeke b, or by proportion, as 
Q2c—b : 2c-+b :: 6 : w : that is, only increase the value of 
x, found by prob. 5, in the ratio of 2c— 6 to 2c+-6. 

Now, in the first example to that prob. the value of x or 
b was there found = 2955 ; and 2c being = 108000, theref. 
2c — b = 105045, and 2c + 6 = 110955, then as 105045 : 
#10955 : : 2955 : 3121 = the value of the height 2 in this 
case, being only 166 feet, or =1,th part more than before. 

Also, for the 2d example to the 5th prob. where 2 was = 
6420 ; ‘therefore as 2c — 6: 2¢ + 6 or. as 105045 : 119055: 
6420: 6780 the height ascended in this example, being ae 
the 18th part more than before. And so on, for any other ex- 
amples ; the value of 2c being the constant number 108000. 


PROBLEM VIHI. 


To determine the Time of a Ball’s Ascending , considering the 
Decreasing Density of the Air as in the last prob. 


The fluxion of the time is = x . But the general equa- 
4) és 
VOL. Eh 60 tan 


466 THEORY AND PRACTICE 


tion of es fluxions of the space x and se v, in the lasi 





GG —VY pel ie —vy 
prob. was = Bo han ren ; ther. z= Hate eare ie 
zw cx —v : 
a | But x, which is al- 
hence ¢ i Cee tas ut «, which 1 


ways small in respect of c, is nearly =b as determined in the 


b 
last problem ; theref. ae may be substituted for gs with- 


: ; ‘ w cec+b —wv 
out sensible error ; and then 7 becomes = 35 X— > Aaa tae 
Now, this fluxion being to that in prob. 6, in the constant ratio 
of c — btoc + b, their fluents will be also in the same ¢on- 
stant ratio. But. by the last prob. ¢ = 54000, and 6 = 2955 
for the first example i in prob. 5; therefore c—b= 51045, and 
¢-+-h=5695h, also, the time in problem 6 was 9”-91 ; there- 
fore as 51045 : 56955 = : 9” 91: 11/04 for the time in this 
_ ease being 1-13 more than the former, or nearly the 9th part 
~ more ; which is nearly the double, or as the square of the 
difference, in the last prob. in the height ascended. 


PROBLEM IX. 


To determine the circumstances of Space, Time, and Velocity, 
of a Ball Descending through the Atmosphere by its own 
| Weight. 


It is here meant that the balls are at least as heavy as cast 
iron, and therefore their loss of weight in the air insensible ; 
and that their motion commences by their own gravity from a 
state of rest. The first object of enquiry may be, the utmost 
degree of velocity any such ball acquires by thus descending. 
Now it iS manifest that the ball’s motion is commenced, and 
untformly increased, by its own weight, which is its constant 
urging force, being always the same, and producing an equal 
increase of velocity in equal times, excepting for the diminu- 
tion of motion by the air’s resistance. It is also evident that 
this resistance, beginning from nothing, continually increases, 
in some ratio, with the increasing velocity of the ball. Now, 
as the urging force is constantly the same, and the resisting 
force always increasing, it must happen that the latter will at 
length become equal to the former*: when this happens, 
there can afterwards be no further acceleration of the motion, 
the impelling force and the resistance being equal, and the ball 
must ever after descend with a uniform motion. It follows 





* This reasoning is not conclusive. The velocity of the descending body in- 
creases. continually, but never becomes equal to a certain determinate velocity. 


therefore 


OF GUNNERY. 467 


therefore that, to answer the first enquiry, we have only to 
determine when or what velocity of the ball will cause a re- 
sistance just equal to its own weight. 


Now, by inspecting the tables of resistances preceding prob. 
1, particularly the first of the three tables, the weight of the 
ball being 1-05lb. we perceive that the resistance increases 
in the 2d column, till 0°69 opposite to 200 velocity, and 1°56 
answering to 300 velocity, between which two the proposed 
resistance 1-05, and the correspondent velocity, fall. _ But, in 
two velocities not greatly different, the resistances are very 
nearly proportional to the squares of the velocities. There- 
fore, having given the velocity 200 answering to the resist- 
ance 0°69, to find the velocity answering to the resistance 
1-05, we must say, as 0°69 : 1:05: : 2002 : v? =60870, theref. 
v= ,/ 60870 = 246, is the greatest velocity this ball can ac- 
quire ; after which it will descend with that velocity uniform- 
ly, or at least with a velocity nearly approaching to 246. 

The same greatest or uniform velucity will also be directly 
found from the rule -00001725v? == r, near the end of prob- 
lem 2, where r is the resistance to the velocity v, by making 


1-05 
1° Et Ve 2 — 
05=r ; for then v?= -00001725 


= 60870, the same value 
for v? as before. } 

But now, for any other weight of ball; as the weights of 
the balls increase as the cubes of their diameters, and their 
resistances, being as the surfaces, increase only as the squares 
of the same, which is one power less ; and the resistances 
being also in this case, as the squares of the velocities, we 
must therefore increase the squares of the velocity in the 
ratio of the diameters of the balls ; that is, as 1-965 :d:: 

2462 d 
= it EE =v? awe eee ees ee 1 
246 *ToG5 * v? and hence v=246,/ 1-965 1753 4/f d. 

If we take here the 3lb ball belonging to the 2d table of 
resistances, whose diameter d is, =2-80 ; then ,/2°80= 1-673, 
and 1751 X 1:67 = 294, is the greatest or uniform velocity, 
with which the 3lb ball will descend. And if we take the 
6lb. ball, whose diameter is 3°53 inches, as in the 3d table of 
resistances: then ,/3°53 = 1°88, and 175} X 1:88 = 330, 
being the greatest velocity that can be acquired by the 6]b 
ball, and with which it will afterwards uniformly descend. 
For a 9lb ball, whose diameter is 4°04, the velocity will be 
1753 X 2:01 = 353. And so on for any other size of iron 
ball, as in the following table. Where the first column con- 


tains the weight of the balls in lbs; the 2d their diame- 
ters 


468 THEORY AND PRACTICE 





ters in inches ; the 3d their Term. [Height 
velocities to which they | Wt. | Diam. ly eioc. ldue to. 
nearly approach, as a limit, Ibs.) inch. feet. |v, feet, 
and therefore called their 

















930 


terminal or last velocities, 1:94 | 244 

with which they afterward 9:45 |: 975 | 1182 
descend uniformly ; and the | 9-80 294 | 1260 
Ath or last column the 3-08 308 | 1482 


1 t 
© 

3 

4 

6 

ties, or the heights from 9.| 4:04 | 353 

12 

18 2450 
24 


1958 

which the balls must descend 4-45 370 | 2139 
in vacuo to acquire them. 5:09 396 

But it is manifest that the 5-60 415 | 2691 

balls can never attain exactly 391-6517 | 436 | 2790 

to these velocities in any 36 1 6-41 444 | 3080 


heights due to these veloci- 3°53 | 330 1701 


finite time or descent, being A2 | 6°75 | 456 | 3940 


only tie limits to which they 
continually approach,without | 
ever really reaching. though they arrive very nearly at them 
in a short space of time; as will appear by the following cal- 
culation. Aarts he toe 

To obtain general expressions for the space descended, and 
the time of the descent, in.terms of the velocity v: put « = 
any space descended, t= its time, and v the velocity ac- 
quired, the weight of the: ballw = 1-05lb. Now, by the 
theorem near the end of prob. 2, which is the proper rule for 
this case, the velocity being small, -00001725v2 = cv? is the 
resistance due to the velocity v; theref. w—cv? is the impelling 











w—cv* : : 
force, and i =f the accelerating force ; conseq. vy or 
3 . ., W—cv? Cb! Oy vv 
Qu fr = Qer X —, and x=— X —, the correct flu- 
w 2g w—cv? 


ent of which, by the 8th form is « = Bact a Na 9 Eee the 
: 4gc w—cv3 
general value of the space x descended. 

Here it appears that the denominator w—cv? decreases ag 
~ Increases ; conseq. the whole value of x, the descent, in- 
creases with v, till it becomes infinite, when the resistance 
cv? 1s = w the weight of the ball, when the motion becomes 
uniform as before remarked. We may however easily assign 
the value of x a little before the velocity becomes uniform, 
er before cv? becomes = w. ‘Thus, when cv? = w, then 
~ == 246, as found in the beginning of this problem. Assume 
therefore v a little less than that greatest velocity, as for in- 
stance 240: then this value of v substituted in the general 


formula 


OF GUNNERY. . 469 


formula for x above deduced, gives x = 2781 feet a little be- 
fore the motion becomes uniform, or when the velocity has 
arrived at 240, its maximum being 246. 

In like manner is the space to be computed that will be 
due to any other velocity less than the greatest or terminal 
velocity. On the contrary, to find the velocity due to any 
® 





' Ww 
ds from the fi lag==—-Xh.l. —. 
proposed space z, ir e formula x Dapk x iain a 
eee f Agcer % 
Here z is given, to find v. . First then ey Fe pt Rte 
. Ww Wi— Cy? 


take therefore the number to the hyp. log. of 





Agca 
5 , which 
w 
w 
number call.w ; then n = ley?) COnseq. Nw — Nev? = w, 
Y—C ; 


? 
a2 


and nw — w= Ne 





for the value of » due to any distance x. Suppose, for in- 
stance, x is 1000. Now 4g being = 64, w = 1:05, and 


4ec 
¢ = 00001725; theref. = 1-0514, and the natural 
number belonging to this, considered as an byp. log. is 
—I1 
2-8617=n ; hence then v = 4/ “aw = 199, is the velocity 


due to the space 1000, or when the ball has descended 1000 
feet. 


Again, for the time ¢ of descent: Hare ,=% ; but 
ve) 








oe = pm cs —, as found above, theref. ¢ = Re Rirath Sine 
2g w—cv* 2g w— cv? 
Ve a + 
the fluent of which: is 7 ants thal ar ar the general 
hn —v 


value of the time ¢ for any value of ihre velocity» ; ; anlen 
value of ¢ evidently increases as the denominator 4/ roe ger 
L 


decreases, or as the velocity v increases ; and consequently 
the time is ae when that denominator vanishes, which 


is when.v == y= us or cv? = w,. the resistance equal to the 


ball’s weight, eiitg the same case as when the space x be- 
comes infinite, as above remarked. But, like as was done. 
for the distance x as above, we may here also find the value 

of 


470 THEORY AND PRACTICE 


of ¢ corresponding to any value of v, less than its maximum 
246, and consequently to any value of 2, as when v 1s 240 for 
instance, or « = 2781, as determined above. Now, by sub- 
stituting 240 for v, in the general formula. . 


Vv; <4 | 
=p J/- ~x h. |. — , it brings out t=16"-575 ; so that 


orem 


it would be nearly 161 seconds when the velocity. arrives at 
240, ora little less than the maximum or uniform degree, viz. 
246, or when the space descended is 2781 feet. 

Also, to determine the time corresponding to the same, or 
when the descent is 1000 feet, or the velocity 199: find the 
W 1 1:05 __ 246 123 

= Then 


1 

lue of —./ — = — = =. 
value ol 4g” ¢. _.64¥ 00001725 64. 32 

an 
Vet” 246-4199 445 
py 5 ~ 946—199 47’ 
Ve 

ISS er ; 

Hence 2°2479 Aa Vata ihe the time of descending 1000 feet, 
or, when the velocity is 199. 

See other speculations on this problem, in Prob. 22, Pro- 
jectiles, as determined from theory, viz. without using the 
experimented resistance of the air. 


the byp. log of which is 22479. 


PROBLEM X. 


To determine the Circumstances of the Motion of a Ball pro- 
jected Horizontally in the Air ; abstracted from its Vertical 
Descent by its Gravitation. 


. Putting d for the diameter, and w the weight of the ball, 
v the velocity of projection, and » the velocity of the ball 
after having moved through the space x. Then by corol. 1 
to prob. 2, if the velocity is considerable, such as usual in 
practice, the resistance of the ball moving with the velocity 


S Mv2—Nnv _ : 
v, 18 (nv? —nv) d?, and therefore ——__—d? is the retardive 
W 
force f; hence the common formula vv = 2gfx, is —vy = 


30x ——“"d?, and theref. x= Loa gL Co), 


32d2 > mv2—nv © 32d? 





es w —v Seerteh : 
Br acs TEP ommeiys the fluent of which is maida’ . 


17% Ww 


OF GUNNERY. ATk 


—~X—hyp. log. of v——, and by the correction by the 





Gado 
nm 
Ww . Ler m 
first velocity v, it becomes Oso aE Xx h. log. —s the 
re) es 
m™m 
general formula for the distance passed over in terms af the 


velocity. 

Now, for an application, let it be required first, to aoe 
mine in what space a 24lb ball will have its velocity reduced 
from 1780 feet to 1500, that is losing 280 feet of its first 
velpcuyy Here, d= 5: Be w = 24, v = 1780, and v=1500; 





also — = 150. Hence - rT: a= = 3587: 4, then ¢ = 3587°:4 X¥ 
—150 630 163 
é _=3 ak ala 
b. 1. 3587-4 X he To = =e 35874 xh}. 122 6°" 


676 feet, the space passed over. when the ball has lost 280 feet 
ef its motion. 

Agaia to find with what velocity the same ball will move, 
after having described 1000 feet in its fiaht The above, 





— 150 
theorem is x or 1000 = 3587-4 X h.1. a == 3587-4 X 
1630 Bets 1630 
h. 1. —— 1, ——==3 
Sobenierrrns ities 71 =h.] SO? but the number to the 
10000 . 1630 
hyp. log. 35874 8 1:7416 = Nn suppose ; then Seeeerayt and 


nv-— 150N= 16360, or nv==1630-+150n, and are — 150 
= 936 — 150 = 786, the velocity, when the ball has moved 
1000 feet. 

Next, to find a theor. for the time of describing any space, 














dest locity: H Oe aes 
er destroying any velogt y: Hereg = -= Say : : 
om 
the fluent of which, by the 9th form is ¢= iT SEN 
5 32md? ” 
Uv 43) a 
h.1. nee ok | i 
a BBnd2 xh! ; et by correction 
m am 
mo x @ |. ———_—h. 1. - ago ar 32n oii qa * hyp. log. 


yeu! 50 


472 THEORS: AND PRACTICE 


v—150v 
v—150 v 
eas: as before. 

Now, to take for an example the same 24Tb ball, and its 
projected velocity 1780, as before; let it be required to find 
in what time this velocity will be reduced to 786. Here then 
v=1780, v = 786, w = 24,d= 56, fe = 31°36, n=-:001 ; 


w 750 - 150 v 1630 
hence - = = 23°916; and ~ SEG wea = ese 


32nd? ‘31-36 
the hyp. log. of which is -1099; then 31:36 


786 _ 1353 
1780 18868’ 
x*1099=2”628, the time required. 

For another example, let it be required to find when the 
velocity will be reduced to 1000, or 780 destroyed. Here 
v==1000, and all the other quantities as before. Then 
v—150 v _ 1630 1000 1630 


Y= 150. ¥. 2850 1780 1513” 
is ‘07449 ; theref. 31:36 X07449=1"-78, is the time sought. 

On the other hand, if it be required to find what will be 
the velocity after the ball has been in motion during any given 
time, as suppose 2 seconds, we eae reverse the calculation 


—, putting v for the first velocity, and 150 for — — its 








the hyp. log. of which 





STEM ay shoe ue —-150 2 : 
thus: ¢=2" being Send” : Sain, = 23-916 X 
v—150 v 
; f. —_— =-083626 is th ; i 
Siar Freie a ; there Sate 083626 is the hyp. log. of 
— 150 
- Tape |» the number cups: to which is 1:08725 =n 
v~150 0. 
$ th = ——-—_ ,-. a = 
suppose, that is, Nn ear pite Hence nvv — 150 nv 
150Nv _—- 290290 


~ 1500, andy =———— avs = 951, the ve- 
Me aes ha aati oA OPENS oe a 10 SEO ha 


locity at the end of 2 seconds. 

The foregoing calculations serve only for the higher velo- 
cities, such as exceed 200 or 300 feet per second of time. 
But, for those that are below 300, the rule is simpler, as the 
resistance is then, by cor. 2, prob. 2, -00000447d? v2 =cd? v2 , 
where d denotes the diameter of any ball. Hence then, em- 








d3y2 et ay 
ploying the same notation as before, oo = sd and — vy = 





aia sx ~, the correct flu- 


ent of males isfi= ois xh. > 


Now, 


OF GUNNERY. 473 


Now, for an example, suppose the first velocity to be 300 
=v, and the last v=100 for a 24\b ball. oe w= 24,d= 


3 
5°6, d?= 31: - c= -00000447 ; therefore ——, as jonaae 
300 i 
5350 ; and © = = = 3, the hyp. loz. of which is 1°0986 ; 


theref. 1: 0986 i Sea Laae ey is the distance.—If the first 
velocity be only 200=v; then “=2, the hyp. log. of which 


is *69315, therefore, 69315 X5350=3708= 7, the distance. 
And conversely, to find what velocity will remain after 
passing over any space, as 4000 an the first velocity being 


v= 200. Here the hyp. log. of ~ cal Aland daha 





— —— «so ———. 


* 5350 5350 535 


8 
= Bees 74766, ‘the natural number of which is 2° 1120, that 
Vv 200 
is, 2412 = = ; therefore v= ee thee spt) the vale» 
city. 
Again, for the time ¢: since tae x —~, therefore 7 
x w 
Sie aor ae 2 the correct fluent of which is ee 7a 
"4 
x (oar) = ears x ane —So, for one if v=300, and 
2 v=—v 900. 2 
#~ =100; then —— ai ecaah S002 then Tye so or 5350 x 
9 


500 =35" 2—t, the time of reducing the 300 velocity to 100, 


or of passing over the space 5878 feet. 
And reversing, to find the velocity'v, answering to any given 


i Si ee ee ini Ta — RTS er ng | SVE 2 
Ging t: Since 1 =35 5, X (; tat, 5350 x (3 0 theref. 
ened Here, if tbe iven=30", and v=300; thenv= 
~~ 5350-Ft0" re te ; ay 
53500 535 - 32100 __ edie 
5350-9000 ~ 1435 => X300== 553 —._—== 112, the velocity sought. 
TOD v 
C l. T ‘ pee] EY 6 j a= 
oro he same form of theorem, x leryar X hl Pe 


as above, is brought out for small velocities, will also serve 
for the higher ones, if we employ the medium resistance es 
tween the two proposed velocities, as was done in prob. 5 

Vor. I. 61 Thus, 


474 THEORY AND PRACTICE 


Thus, as in the first example of this problem, where the two 
velocities are i78U and 150, the resistance due to the velo- 
city 1700, in the first table of resistances, being 74°13, say as 
17002 : 17802 : : 74°13 : 81-27, the resistance due to the ve- 
locity 1780; then the mean between 81°27 and 57:25, due 
to 1800 velocity, is 69-26, or rather take 691. Again, as 
s/f 65'7 : 4/ 694 : : 1600: 1646, the velocity due to the me- 
dium resistance 691. Hence, asin prob. 5, as 16462 : v7 :: 
691 : -00002565v? = suppose av?, the resistance due to any 
velocity v, between 1780 and 1590, for the 1-05lb ball. And 
as 1°9652 : 5:6? 2; av® :8'124av? == -00020838v? =—bv? sup- 
pose, the resistance due to the same velocity, with the 24lb 


» aiiby? ; ¥ 7 | 


. = 3v eae ae S 
Ear Oe the correct fluent of which is mr x hi}. 54a 
Mie 3 89 ie 


velocity sought. 
PROBLEM XI. 


To determine the Ranges of Projectiles in the ir. 


To determine by theory, the trajectory a projectile de- 
scribes in the air, is one of the most difficult problems inthe 
whole course of dynamics, even when assisted by all the ex- 
periments that have hitherto been made on this branch of the 
physics ; and is indeed much too difficult for this place, in the 
full extent of the problem : the consideration of it must there- 
fore bé reserved for another occasion when the nature of the 
air’s resistance can be more amply discussed. Eventhe solu- 
tions of Newton, of Bernoulli, of Euler, of Borda,. &c. &c. 
after the most elaborate investigations, assisted by all the ree 
sources of the modern analysis, amount to no more than dis- 
tant approximations, that are rendered nearly useless, even to 
the speculative philosopher, from the assumption of a very 
erroneous Jaw of resistance in the air, and much more so to 
the practical artillerist, both on that account, and from the 
very intricate process of calculation, which is quite inapplica- 
ble to actual service. The solution of this problem requires, | 
as an indispensible datum. the perfect determination by ex- 
periment of the nature and laws of the air’s resistance at dif- 
ferent altitudes, to balls of different sizes and densities, mov- 
ing with all the usual degrees of celerity. Unfortunately how- 
ever, hardly any experiments of this kind have been made ex- 
cepting those. which on some occasions have been published 

by 


OF GUNNERY. 475 


by myself, as in my tracts of 1786, as well as in my Diction- 
ary, some few of which are also given in art. 105 of Mot. and 
Forces, with some practical inferences, And though I have 
many more yet to publish, of the same kind, much more ex- 
tensive and varied, | cannot yet undertake to pronounce that 
they are fully adequate to the purpose in hand. 

All that can be here done then, in the solution of the pre- 
sent problem, besides what is delivered in this volume, is 
to collect together some of the best eres rules, founded 
partly on theory, and partly en practice. 1, In the first place 
then, it mee be remarked, that the initial or first velocity of 
a ball may be directly computed by prob. 17, page 393 of 
this volume ;, having given the dimensions of the piece, the 
weight of the ball, and the charge of powder. Or otherwise, 
the same may be made out from the table of experimented 
ranges and velocities in page 141 of this volume, by this rule, 
that the velocities to different balls. and different charges of 
powder, are as the square roots of the weights of the powder 
directly, and as the square roots of the weights of the balls 
inversely. Thus, if it be enquired, with what velocity a 24lb 
ball will be discharged by 8lb of powder. Now it appears in 
the table, that 8 ounces of powder discharge the 1lb ball with 
1640 feet velocity; and pecan &lb are = 128 ounces ; 
therefore by the rule, as + 2 128 :: 1640: 1640 ,/ 18 

= 1640 ¥ % = 1339, the velocite sought. Or otherwise, by 
rule 1, p. ‘142 of this vol. as 4/ 24:4/ 16: : 1600: 1306, the 
same velocity nearly. But when the charges bear the same 
ratio to one another as the weight of the balls, that is when 
the pieces are said to be alike “charged. then the velocities 
will be equal. Thus, the ilb ball by the 2 oz charge being 
the 8th part of the weight, and the 24lb ball, with 3lb of pow- 
der, its 8th part also, will have the same velocity, viz. 860 
feet. In like manner, the 1230 tabular velocity, answering 
to 4 oz of powder, the 4th part of the ball, will equally belong 
to the 24lb ball with 6lb of powder, being its 4th part and the 
Bpalae velocity 1640, answering to the 80z charge, which is 

1 the weight of ball, will equally belong to the 241b ball with 
12tb of powder, being also the 1 of its weight. 

2. By prob. 9 will “he found what is called the terminal velo- 
city, that is, the greatest velocity a ball can acquire by des- 
cending in the air; indeed a table is there givgn of the seve- 
ral terminal velocities belonging to the coals with the 
heights. in an annexed column, due to those velocities in va- 
cuo, that is the heights from which a body must fall in vacuo, 
to acquire those velocities. 

3. Given the initial velocity, to find the elevation of the 

/ plece 


476 THEORY AND PRACTICE 


piece to have the greatest range, and the extent of that range. 
These will be found by means of the annexed table, altered 
from): Professor... Robisen’§, ce jis a 
in the Encyclopedia Bri- |Lable of Elevations giving the 
tannica, and founded on an ee aes hes i 
approximation of Sir I. |!mtialvel. : ANSE PIV 
Rew ton's. The numbers |div. by v. peeral by a, 
in the first column, multi- |—————— 
plied by the terminal velo- | 06910 | 44° O' | 0-3914 
city of the ball. give the | 9°9445 | 45 15 | 0°5850 
initial velocity; and the | 1:1980 | 42 80 | 0°7787 
numbers inthe last co- | 1°4515 | 41 45 | 0:9724 
lumn, being multiplied by | 1°7050 | 41 0 1°1661 
the height, give the great- | 119585 | 40 15 | 1°3598 
est ranges ; the middle co- | 2°2120 | 39 30 | 1:5535 
lumn showing the elevations | 2°4655 | 38 45 17472 
to produce those ranges. 2°7190 | 38 O | 1:9409 
To use this table then, | 2°9725 | 37 15 2°1346 | 
divide the given initial ve- | 3°2260 | 36 350 | 2°3283 
locity by the terminal ve- | 3°4795 | 95 45 | 2°5220 | 
locity peculiar to the ball, | 3°7930 | 35 0 | 2°7157 
found in the table in ‘prob. | 3°9865 | 34 15. 1 2°9094 | 
9, and look for the quo- | 4°2100 | 33 30 | 3:1031 
tient in the first column | 4°4935 | 32 45 | 3°2968 | 
| 











here annexed. Against this, | 4°7470 | 32 0 | 3°4905 
in the 2d column will be |_9°0000 
found the elevation to give 
the greatest range ; and the number in the 3d column multi- 
plied by.a, the altitude due to the terminal velocity, also 
found in the table in problem 9, will give the range, nearly. ~ 
Ex. 1. Let it be required to find the greatest range of a 
24\b ball, when discharged with 1640 feet velocity, and the 
corresponding angle to produce that range. By the table in 
prob. 9, the terminal velocity of the 24lb ball is 415, and its 








: Z 1640 
producing altitude 2691: hence Tie 


3°9865 in the first column of our table, to which corresponds 
the angle 34° 15’, being the elevation to produce the greatest 
range ; and the corresponding number 2.9094 in the 3d co- 
lumn, multiplied by 2691’, gives 7829 feet, for the greatest 
range, being nearly a mile and a half. 
_ Exam, 2. In like manner, the same balls discharged with 
the velocity 860 feet, will have for its greatest range 3891 
feet, or nearly 3 of a mile, and the elevation producing it 
39° Ob). 
These examples, and indeed the whole table in the 9th 
problem, 


= 3°95, nearly equal to 


OF GUNNERY. Avy 


problem, are only adapted to the use of cannon balls. But 
it is not usual and indeed not easily practicable, to discharge 
cannon shot at such elevations, in the British service, that 
practice being the peculiar office of mortar shells. On this 
account then it will be.necessary to make out a table of ter- 
minal velocities, and altitudes due to them, for the different 
sizes of such shells. The several kinds of these in present 
use, are denominated, from the diameters of their mortar 
bores in inches, being the five following, viz. the 4-6, the 
5:8, the 8, the 10, and the 13 inch mortars, as in the first 
column of the following table. But the outer diameters of 
the shells are somewhat smaller, -to leave a little room or 
space as windage, as contained in the 2d column. 


Table of dimensions, &c. of Mortar Shells. 



































: Weight | Weight |Ratio of|Termi-|Alt. a due 
Saag i off Shellsf equal|Shell elaed ve-|to veloci- 
I " |filled. jsolid. jsolid. — {locity. Kes 
inch. | inch. lbs. Ibs. feet. feet. 
4:6 4°53) 9 123 1°42 314 1541 
58 | 572 1° 18 | 251 | 1-42] 352 | 1936 
8 7°90 47 | 67 1°42 | 414 | 2678 
10 9-84 911 | 130 1-42 462 3335 
13 (12°80 201 286 1°42 527 | 4340 





The 3d column contains the weight of each shell when the 
hollow part is filled with powder : the diameter of the hollow 
is usually ;4, of that of the mortar: the weight of the shells 
empty and when filled, with other circumstances, may be seen 
at quest. 53, in Mensuration, end of vol. 1. On account of the 
vacuity of the shell being filled only with the gunpowder, the 
weight of the whole so filled, and contained in column 3, is 
much less than the weight of the same size of solid iron, and 
the corresponding weights of such equal solid balls are con- 
tained in col. 4. The ratio of these weights, or the latter di- 
vided by the former, occupies the 5th column. 

Now because the loaded or filled shells are of less specific 
gravity, or less heavy, than the equal solid iron balls, in the 
ratio of 1 to 1°42, as in column 5, the former will have less 
power or force to oppose the resistance of the air, in that 
same proportion, and the terminal or greatest velocity, as 
determined in the 9th prob. will be correspondently less. 
Therefore, instead of the rule there given, viz. 175:5,/d, for 


that velocity, the rule must now be 1755/5 = 147°3,/d=0 
| the 


478 THEORY AND PRACTICE 


the diameter. of the shell being d ; that is, the terminal velo- 
cities will be all less in the ratio of 147-3 to 175°5. Now, 
computing these several velocities by this rule, to all the dif- 
ferent diameters, they are found as placed in the 6th col. ; 
and in the 7th or last column are set the altitude which would 
produce these velocities in vacuo, as computed from this 
> UV 
theorem aa" 
Having now obtained these terminal velocities, and their 
producing altitudes, for the shells, we can, from them and the 
former table of ranges and elevations, easily compute the 
greatest range, and the corresponding angle of elevation, for 
any mortar and shell, in the same way as was done for the 
balls in this problem. Thus, for example, to find the great- 
est range and elevation, for the 13 inch shell, when projected 
with the velocity of 2000 feet per second, being nearly the 
greatest velocity that balls can be discharged with.. Now, 
9 
by the method before used it = 3°796; opposite to this, 
found in the first column of the table of ranges, corresponds 
34° 49’ for the elevation in the 2d column, and the number 
2-764 in the 3d column ; this multiplied by the altitude 4340, 
sives. 11995 feet, or more than 2} miles, for the greatest 
range. 

This however is much short of the distance which it is said 
the French have lately thrown some shells at the siege of 
Cadiz. viz. 3 miles, which it seems has been effected by 
means of a peculiar piece of ordnance, and by loading or fill- 
ing the cavity ef the shell with lead to render it heavier, and 
thus make it fitter to overcome the resistance of the air. Let 
us then examine what will be the greatest range of our 13 
inch shell, if its usual cavity be quite filled with lead when 
discharged, with the projectile velocity of 2000 feet. 

Now the diameter of the cavity, being about 4, of that of 
the mortar 13, will be nearly 9 inches. And the weight of 
a globe of lead of this diameter is 139°3lb ; which added to 
187-8, the weight of the shell empty, gives 327lb, the whole 
weight of the shell when the cavity is filled with lead, which 
was found 286 when supposed all of solid iron, their ratio or 
quotient is -8783. Then, as before, the theorem will be 


175°5 / ease 187-34/ d for the terminal velocity ;_ which, 3 
when d = 12°8, becomes 670 for the terminal velocity ; 
We eas 5 6708 oe 
therefore its producing altitude is ae i014. Then, by 
i the 


OF GUNNERY. 479 


2000 


670 
found in the first column of the table of ranges, the opposite 
number in the 2d col. is 37° 15’ for the elevation of the piece, 
and in the 3d column 2°14, multiplied by 7014, gives 15010 
feet, or nearly 3miles. So that our 13 inch shells, discharg- 
ed at an elevation of about 371 degrees, would range nearly 
the distance mentioned by the French, when filled with lead, 
if they can be projected with so much as 2000 feet velocity, 
or upwards. This however it is thought cannot possibly be 
effected by our mortars ; and that it is therefore probable the 
French, to give such a velocity to those shells, must have 
contrived some new kind of large cannon on the occasion. 

4. Having shown in the preceding articles and problems, 
how, from our theory of the air’s resistance, can be found, 
first the initial or projectile velocity of shot and shells; 2dly, 
the terminal velocity, or the greatest velocity a ball can ac- 
quire by descending by its own weight in the air; 3dly, the 
height a ball will ascend to in the air, being projected verti- 
cally with a given velocity, also the time of that ascent ; 4thly, 
the greatest horizontal ranges of given shot, projected with a 
given velocity ; as also the particular angle of elevation of 
the piece, to produce that greatest range. It remains then 
now to inquire, what laws and regulations can be given re- 
specting the ranges, and times of flight, of projects made at 
other angles of elevation. on 

Relating to this inquiry, the Encyclopedia Britannica men- 
tions the two following rules: ist ‘* Balls of equal density 
projected with the same elevation, and with velocities which 
are as the square roots of their diameters, will describe similar 
curves. This is evident, because in this case, the resistance 
will be in the ratio of their quantities of motion; therefore all 
the homologous lines of the motion will be in the proportion 
of the diameters.” But though this may be nearly correct, 
yet it can hardly ever be of any use in practice, since it is 
usual and proper to project small balls, not with a less, but. 
with agreater velocity, than the larger ones. 2dly, The other 
rule is, ‘‘ If the initial velocities of balls, projected with the 
same elevation, be in the inverse subduplicate ratio of the 
whole resistance the ranges, and all the homologous lines in 
their track, will be inversely as those resistances.’’ This rule 
_will come to the same thing, as having the initial velocities in 
the inverse ratio of the diameters, as distant perhaps from fit- 
ness as the former. Two tables are next given in the same 
place, for the comparison of ranges and projectile velocities, 
the numbers in which appear to be much wide of the truth, 
as 


the same method as before, = 2-985; which number 


480 PROMISCUOUS EXERCISES. 


as depending on very erroneous effects of the resistance. 
Most of the accompanying remarks, however, are very inge- 
nious, judicious, and philosophical, and very justly recom- 
mending the making and recording of good experiments on 
the ranges and times. of flight of projects, of various sizes, 
made with different velocities, and at various angles of eleva- 
tion. 

Besides the above, we tind rules laid down by Mr. Robins 
and Mr. Simpson, for computing the circumstances relating to 
projectiles as affected by the resistance of the air. Those of 
the former respectable author, in his ingenious fracts on 
Gunnery, being founded on a quantity which he calls r, (an- 
swering to our letter a in the foregoing pages), I find to be 
almost uniformly double of what it. ought to be, owing to his 
improper measures of the air’s resistance ; and therefore the 
conclusions derived by means of those rules must needs be 
very erroneous. Those of the very ingenious Mr. Simpson, 
contained in his Select Exercises, being partly founded on 
experiment, may bring out conclusions in some of the cases 
not very incorrect ; while some of them. particularly those 
relating to the impetus and the time of. flight, must be very 
wide of the truth. We must therefore refer the student, for 
more satisfaction to our rules and examples before given in 
page 142 this vol. &c. especially for the circumstances of dif- 
ferent ranges and elevations, &c. after having determined, as 
above, those for the greatest ranges, founded « on the real mea- 
sure of the resistances. 


oe pees 


PROMISCUOUS PROBLEMS, AS EXERCISES IN ME- 
CHANICS, STATICS, DYNAMICS, HYDROSTATICS, 
HYDRAULICS, PROJECTILES, &c. &c. 


PROBLEM I. 


Let az and ac be two inclined planes, whose common altitude an 
is given = 64 feet ; and their lengths such that a heavy body 
is 2 seconds of time longer in descending through ap than 
through ac, by the force of gravity ; and if two balls, the one 
weighing 3 and the other. 2lb, be connected by a thread and 
- laid on the planes, the thread ‘sliding freely over the vertex A, 
- they will mutually sustain each other. Quere the lengths of 
the two planes. 


Tue lengths of the planes of the same height being as the 
times of pig down them (art, 133 this vol.), and also ho 
the 


PROMISCUOUS EXERCISES. 481 


the weights of bodies mutually sustaining each other on them 
(art, 122), therefore the times must be as the weights ; hence 
as 1, the, difference of the. weights, is to 2 sec. the dif. of 


times, : ie f at : tis the times of descending down the two 


planes. And as 4/ 16: J 64:; 1 sec, : 2 sec. the time of 
descent down the perpendicular heer ¢ art 70.) Then by the 
laws of descents (art. 132), as 2 sec, : 64 feet et} : ae 120 § 
feet, the lengths of the planes. 

Note, In-this solution we have. ‘setae 16 fegt’ as the 
space freely descended by bodies in the Ist second of time, 
and 32 feet as the velocity acquired in that time, omitting the 
fractions 1, and 1, to render the numeral calculations sim- 
pler as was done i in the preceding chapter on projectiles, and 
as wé shall do also in solving the following: questions, yeaah: 
ever. such numbers occur. ; 


Another Solution by means of Algebra. 


Put « = the time of descent down the less plane ; then 
willz + 2 be that of the greater, by the question. Now 
the weights being as. the lengths of the planes, and these 
again as the times, therefore as 2:3::2:2 + 2; hence 
2a-+-4 = 3x, and «= 4sec.. Then the lengths of the planes 
are found as in the last proportion of the former solution. 


i PROBLEM 2. 


if an arp ball fall from the height of 50 feet above the lave 

of the horizon, ‘and impinge on. the hard surface of @ plane 

-anclined to it in an angle of 15 degrees ; itis required to find 

what part of the plane it must strike, so that after reflection, it 

- may fall on the horizontal plane, at the greatest distance Hopes 
‘ble beyond the bottom of the inclined plane! ee 


| Here it is manifest that as 
‘the bali must strike the ob- 
lique plane continued on a 
point somewhere below the 
horizontal plane ; for other- 
‘wise there could be no max- 
imum. Therefore let sc be , : 
the inclined. plane, cpg the horiebbtak one, B the point on 
which the ball impinges after falling from the point a, Beat 
the parabolic path, « its vertex, 8H a tangent at B, being the 
direction in which the ball is reflected ; and the other lines as 
are evident in the figure. Now, by the laws of reflection, 
the angle of incidence ABC, is equal to the angle of reflection 
Vou. Al “69 /HBM, 





482 SIRSOE US EXERCISES. 


“pM, and fevefbie’ this latter, as well as the former, is equal 
to the complement of the 4 c, the inclination of the two planes ; ;: 
but the part 1m is = Zc, therefore the angle of projection 
HBI is = the comp. of double the Zc, and being the comp. of 
wex, theref. Zunk=22Zc¢. Now, put a=50=—ap the height 
above the horizontal line, f==tang. Z prc or 75° the comple- 
ment of the plane’ s inclination, s=tang. HBI or ZH=60° the 
comp. of 2c, s=sine of 2/7 HB1=120° the double elevation, 

or. = sine of 4/c; also x = xB the impetus or height fallen 
through. Then, 

BI = 4KH=2sx, by the prejectilds’ prop. QI, 
ana ; BK = t XKH=1s7x 

cp =t Xsp=t (2 — —a). 
also, KD = BK — Bp = jer — 2 + a, and xe = LBl = sz 5 


then, by the parabola, ,/ BK ne DK : pe : FG = KE X of 
3 Lied 
4 32 2 43 +2usr 2 [se ax (== 92) 2] 52b 


(ax —b?a?), putting b = sine of 2 Zc=sine of 30°. Hence 
co=cp-+pr-tbre=te -ta+set 2b,/ (ax b?2?2) a maximum, 
the fluxion of which made Lets, and the equation reduced, 


by trigonometry ; 











gives a=, X (Ic 0 V (nt raat where n=s +t, and the 


double sign 4 answers to the two roots or values of x, or to 
the two ‘points c, c, where the parabolic path cuts the hori- 
zontal line cc, the one in ascending and the other in descend- 
ing. : 

Now, in the present case, when the Lo 15°, t= tang. 
759°=2 + 4/3, c= tan 60° =,/3, s = sin. 60° = 3 4/3, b= 


sin. 309 es n=stia2+} $33 ; then ae == 100, and 

n2 ak hs _41+6,/3. n2 7 

2 --4b4 Pe is 52 ao 
41+6,/3 

==100X (14 as —~ me hace 100% (1: 99414) = 199° 414 or 


°586; but the Bone: must be taken. Hence the body must 
strike the inclined plane at 149°414 feet below the horizontal 
_ line ;. and its path after reflection i! cut the said line i in two 


; theref. tap X(It 


points ; ; or it will touch it when sx = 5 


est distance cc required i is 826- 9915 feet. 

- Corol. If it were required to find ce or tz-—ta + sat 2b,/ 
(ar—b? a? )=g a given quantity, this equation would give the 
valde of © ‘by pobving: a Tesiretic 


Hence also. the great- 


fie , 


“PROBLEM 


PROMISCUOUS EXERCISES. 438 


_PROBLEM 5 


Suppose w rsh to sail Past the Orkney Islands, in latitude 'B9°B! 
north, om a.N.N. B. course, at the rate of 10 miles an hour ; 
at as required to determine how. long it will be before she ar- 
rives.at the pole, the distance she Hi have sailed, and the oe 
ference ef longitude she will have made: when she. arrives Brera? 


Let arc represent part of the equator ; 
pr the pole ; amrp a loxodromic or rhumb 
line, or the path of the ship continued to 
the equator; PB,’ pc, any two meridians in- 
definitely near enh. other; nr, or mt, the _ 
partofa wba of latitude intercepted be-" 
tween them. 

Put c forthe cosine, yandt fae the tangent 
of the course, or angle nmr to the radius r 3 | 
am, any variable part of the rhumb from the equator, 7 
the latitude sm = w; its sine x, and cosine y; and as, the 
dif. of longitude from a, = z.. Then, since the ‘elementary 
triangle mnr may be considered as a feat: angled plane tri- 
aBEIE it is, as rad. r:¢ =sin. Zmrn:: v= mr: w=mn 


Uw; theref. o=rw, or v= ——— by putting 8 for 
1 2 





the secant of the Z.umr the ship’s course. In like man- 
ner, if w be any other latitude, and v its corresponding length 
WwW 





of the rhumb ; then Aiea age and hence y—v =r X 
ee 


rd 
or D = — , by putting p bp = v—v the diatance) and d = Ww 2 


the dif. of latitude ; which is the common rule. | 

The same is evident without fluxions : for since the Zmrn 
is the same in whateyer point of the path amrp the point mis 
taken, each indefinitely small particle of AMrP, must be to the 
corresponding indefinitely small part of nm, in the constant — 
ratio of radius to the cosine of the course: and therefore the 
- whole lines, or any corresponding parts of them, must be in 
the same ratio also, as above determined. In the same man- 
ner it is proved that radius : sine of the course : : distance ; 
the departure. | Wee , 

» Again, as the radius r : ¢ = tang amr: : w=mn: nr or mt, 
and as r:4y:: PB: pm: : z=BC: mt; ; hence, as the extremes 
_ of these proportions are the same, the rectangles of the means 


: ira 
must be equal, Viz. ye =tyw= en because w= by the pre: 
TOR Silke aa 
perty of the cirele ; thie ef. : we ahh a ae ; the general 
yf mF fluents 


Age secsoamacinits EXERCISES. 


flaents of these are z = ¢ x x hyp. log. vs Mite ns C5. which 


3 corrected by supposing z= 0 when x == a, are 2 =t x x (hyp: : 


log ae — hyp. log vy 





— hyp. log. 4/ Xe is the maritoral parts of the dif. of the 
latitudes oie sines are x and a, which call by then is 2 = 
tb Z 
a the same as it is by Mercator’ s sailing. 


Further, putting m = 2°71828 the number whose fis fo. 





2% pe a 
is 1, andrz= ae nee when z begins at a, m? = is and 


—% 
at | 
theref. er K =r : hence it appears that 


te m4” 


asm”, OF rather. or z increases (since m is conse that a 


approximates to an equality with r, ‘because — wp teotenses 
or converges to 0, which is its limit; cinscquentty r is the 
mit or ultimate value of «: but when x = r, the ship-will 
be-at the pole ; theref.: the pole must be the limit, or eva- 
nescent state, or the rhumb or course: so that the ship may 
be said to arrive at the pole after making an infinite pumiet 


of revolutions round it; for the above expression wen * va- 
nishes when n, and consequently 2, is infinite, in which case x 
is 7. : 
rd sd 

Now, from the equation Che Hamer 4 a: is faites ‘that 
when d = 30° 57’, the comp. of the given lat. 59° 3’, and c= 
sine of 67°, 30’ the comp. of the course, » will be = 2010 
‘geographical miles, the required ultimate distance ; which, 
at the rate of 10 miles an hour, will be passed over in 201 
hours, or 88 days. The dif. of long. is shown above to be in- 
finite. When the ship has made one revolution, she will be 
but about a yard from the pole, considering her as a point. 


» When the ship has arrived infinitely near the pole, s she will © 
go round in the manner of a top, with an infinite velocity ; 
which at once accounts for this paradox, viz. that though she 
make an infinite number of revolutions round the pole, yet 
her distance run Will have an ultimate and definite value, as 
above determined: for it is evident that however great the 
number of revolutions of a top may be, the space passed over - 


by 


PROMISCUOUS EXERCISES. 485 


by its pivot or bottom point, while it continues on or nearly 
on the same point, must be infinitely small or less. than a cer- 
tain assignable quantity. ae 


‘PROBLEM 4, 


A current of water vs discharged by three pee openings or 
slices, in the : following shapes: the first a rectangle, the se- 
cond a semicircle, and the third a parabola, having their al- 
titudes equal and ‘tHeis bases in the same horizontal line, and 
the water level with the tops of the urches: on this supposition 

tt is required to show what may be the Proportion of the quan- 

~ taties discharged by these sluices. + 

~ Let ve be half the parallelogram, avec 

half the semicircle, and avp half the pa- 
rabola, thatis, the halves of the respective 
sluices or gates. Put a= av the common 

altitude, and c = +7854: then is ca? the ?P/-—— 

area of each of the figures ; also ca =AB, A BCD 

a= ac, and 2ca.= ap: also put x = vp 

any variable depth, and =rp. Then the water discharged, 

at any depth x, being as the velocity and aperture, and the 

velocity being in all the figures as ./ x, therefore x ./rx PQ, 


at 
and La/LK PR, and 2a X Ps, or cax?x, and rz 4/ (2a—2), 
and 3¢,/a Xaz, are proportional to the fluxions of the quan- 
tity of water discharged by the said figures or sluices~ re- 
spectively ; the correct fluents of HOKE, when « = a, are 


2 cat, and 2 22 (8 ¥/ 2— 7), and Sca?, the 2d fluent being 
found by art. 60 pa. 336 of this vol. Hence the quantities 
of water discharged by the rectangle, the semicircle, and the: 
parabola, are SPS as 2c, and 2. (8/2—7), and jc, 


or. as 1, and — —— bo (8/2 — Ts and $, or as 1, and 1- 09847, and 
1k 





PROBLEM is; 


‘The ‘ai velocity of. a 241b ball of sas tron, mbiok s 1s project- 
ed in a direction perpendicular to the horizon, being sup- 

. ‘posed 1200 feet per second ; and that the resistance of the me- 
dium is constantly as the square of the velocity, and every 
where of the same density : required he time of Hight, rea 
the height to which it will ascend. — 

- Answer. By problems 5 and 6, of the last chapter, the as« 


- cent will be found = = 5337 feet and the time of the ascent 28 
seconds. 


PROBL PN 


486 PROMISCUOUS EXERCISES. 


PROBLEM 6. 


ay by 


TN dornitie the same as in the last question, supposing the den- 
‘sity of the atmosphere to decrease am ascending after the usu- 
al way? . 

Ans. By probs. 7 and 6 , the height will be 5614 feat, and 

the’ tag 34 seconds. : . | 

- PROBLEM 7. 


itis requrred to fmt the diameter af a circular parachute, by 
means of which a man of 15016 weight may descend on. the 
‘earth, from a balloon at a height in the air, wrth the veloerty 
of only 10 feet in a second. of time, being the velocity acquired 
by a body freely descending through a space of only 1 foot 62 
inches, or of a man jumping down from a height of 183 inches 
the parachute being made of such materials and’ thickness, that 
a circle of it of 50 feet diameter, weighs only 1501b, and so in 
proportion more or less according to the area of the circle. 


If a falling body descend with a uniform velocity, it must 
necessarily meet with a resistance. from the medium it de- 
scends in, equal to. the whole weight that descends. . Let x 
denote the diameter ef the pagent, and a = °7854 ; then 
ax* will be its area, and as 502 : a? : : 150: 3.22 the weight 
of ays same, to which adding ‘eat. the man’s weight, the 
sum 2 r°-+-150 will be the whole descending weight. Again, 
in ae ‘table of resistances (in the scholium to prop. 22, mot. 
of bod. in Fimds), we find that ‘a’ circle of 2 of a square foot 
area, moving with 10 feet velocity, meets with a resistance of 
57 ounces == °0475Ib; and the resistances, with the rane ve- 
locity being as the surfaces, therefore, as 2 : +0475: : ax? : 
*21375ax2 = +16788x2 the resistance of the air to the para- 
eink to which the descending weight must be equal : that 

1678822° = ~a*-+150; hence :10788x? = 150, or x? 
<= - 1390: 5, and hence # = 372 feet, the diameter of re pa- 
rachute ‘required. 

PROBLEM 8. — 


To determine the effects of Pile Engines. 


The form of the pile-engine, as used by the ancients, is not 
known. Many have been invented and described by the 
moderns. Among all these, that appears to be the best which 
was invented by Vauloue, as described by Desaguliers, and 
was used at piling the foundations at building Westminster 
Bridge. Its chief properties are, that the ram or weight be 
raised with the least expense of force, or with the fewest men; 
that it fall areely from its. greatest height ; and that, pe 

allen 


PROMISCUOUS EXERCISES. 487 


fallen, it is: presently laid held of by the forceps, and so raised 
up to its height again, By which means, in the shortest time, 

and with the fewest men, or the least force, the most piles 
can be driven to the greatest depth. © 

Belidor has given some theory as to the effect of the pile- 
engine, but it appears to be founded on an erroneous princi- 
ple: he deduces it from the laws of the collision of bodies. 
But who does not perceive that the rules of collision suppose 
a free motion and a non-resisting medium ? It cannot therefore 
be applied in the present case, where a very great resistance 
is opposed to the pile by the ground. We shall therefore 
here endeavour to explain another theory of this machine. 

‘Since the percussion of the weight acts on the pile during 
the whole time the pile is penetrating and sinking in the 
earth, by each blow of the ram, during which time its whole 
force is spent ; it is manifest that» the effect of the blow is of 
that nature which requires the force of the blow to be esti- 
mated by the square of the velocity. But the square of the 
velocity acquired by the fall of the ram, is as the height it 
falls from; therefore the force of any blow will be as the 
height fallen through. But it is also more or less in propor- 
tion to the weight of the ram; consequently. the. effect or 
force of each blow must be directly in the componnd ratio of 
both, viz. as aw, where w denotes the weight, and a the alti- 
- tude it falls from ; or it will be simply as the altitude ¢, age 
the weight w is constant. 

Again, the force of the blow is opposed by the mass of the 
pile, and by the consistence of the earth penetrated by the 
point of the pile, and also by the friction of the earth against 
the surface or sides of the pile that have penetrated below the 
surface. Consequently the effect of the blow, or the depth 
penetrated by the pile, will be inversely in the compound 
ratio of these three, viz. inversely as mtf, where m denotes 
the mass of the pile, ¢ the tenacity or cohesion of the earth, 
and f the friction of the surface penetrated in the earth. But, 
inthe same sail and with the same pile, m and ¢ are both con- 
_stant, in which case the depth of penetration will be inversely 

only as f the friction. cae all accounts then the penetration 


will be ae a or r simply as ¥ only, for the same weight and 
pile and soil. | | 
To determine the depth sunk by the pile at each stroke of the ram. 


After'a. few ‘strokes, so as to give the pile a little hold in 
the ground, to wen, it stand firmly, the blows of the ram may 
be 


488 PROMISCUOUS EXERCISES. 


be considered as commencing and causing the pile to sink a 
little at every. stroke, by which small successive sinkings uf 
the pile, the space the ram falls through will be successively 
increased by these small accessions, and the force of the suc- 
cessive blows proportionally increased. But these, on the 
other hand, are resisted and opposed by the friction of the 
part of the pile which has been sunk before, and which also 
sinks at each stroke ;. and as the quantities of these rubbing 
surfaces increase in a greater ratio to each other, than the 
heights fallen through, that is, the resisting forces increasing 
faster than the impelling forces, it is manifest. that the depths 
successively sunk by the blows must gradually decrease by 
little and little every time ; which is also found to be quite 
conformable to experience. Thus then the successive sink- 
ings will proceed gradually diminishing, till they become so 
small.as to be almost imperceptible... , | 


Now it was found above that — is as the penetration by any 


blow ofthe ram, by the same pile in the same soil, that is, as 
the height fallen directly, and as the resistance or friction in 
the earth inversely. Let a denote any other and greater 
height, by an after stroke, and F its friction ; also p the pene- 
tration by the former blow, and p that by the latter, which 
must. be the smailer: then, by the foregoing principle, 
a oA 
¥ 

theorem... : : ps “ey 

But now, with respect to the quantity of friction from any 
blow, though it be not known from experiment that the fric- 
tion is exactly proportional to the rubbing surface, there is 
great reason to believe that it must be at least very nearly 
so: there is also equal reason to conclude that the effect 
or resistance from that rubbing surface must be nearly or 
exactly as the length of space it moves over, that is by the 
penetration of the pile by any blow. Now, if d denote the 
depth of the pile in the ground before any new blow is struck 
by the ram, and 6 the depth or penetration produced by the 
blow, then the length of the rubbing surface will be d-+- 1 6; 
for, the length of the rubbing surface is only d at the begin- 
ning of the motion, and it is d-+ b at the end of it, the me- 
dium of the two, or d+ 16, is therefore the due length of 
the surface, and the space or depth it moves over is b ; there- 
fore the whole resistance from the friction is (d + 16)b. Wf 
p then denote any other, depth of the pile in the earth, and 
5 the next penetration, then (p + 4b’) 0 will be its friction. 
Substituting now b for P, and 6° for p, also d+ 1b ey 

D+} 


:: Pip; hencea FA ri fe 3 Fp, which is a general 


PROMISCUOUS EXERGISES, 489 


p-+ 16! for r, in the general theorema:a:: fr: rp, it be- | 
comes a: a:: (d+1b)b: (o-++1b’)b, for the general relation, 
between the heights fallen and the resistance and penetration. 

This theorem will very conveniently give the series of ef- 
fects, or successive sinkings of the piles, by the blows of the 
ram. Thus, after the pile has been properly fixed, or indeed 
driven to any depth in the earth, denoted by d, then to give 
a blow, the ram falls from the height a + d, and thereby 
sinks the pile the space b suppose ; hence for the next stroke, 
the fall will be a-+-d-+6= a in the theorem above, and 
D+ 46=d-+ b+ 40’, the next penetration or sinking being 
6%; theref.a+-d:a+d+6:: (d+ 4b)b: (d-+b-- 1b), 
a proportion which gives the quadratic equa. 6’?-+-2b'(d-++b)= 
ex (2d-+-6)b, the root of which is b'== — (d+ 6) 4- 

rg ttEtb (og 4 py a chete dt 
Vv [(4 +)? + aed X (2d + b)d] pay: X ar, ¢ 
i 

cae? bnearly, because 6 is small in com: 


d-+-b 











nearly, or indeed = 
parison with a-+d. 


Now, for am example in numbers, suppose a = 5 feet = 
60 inches, d= 10, 6 = 3, that is a = 60 the height of the 
ram above the top of the pile before this enters the ground ; 
d = 10, after being fixed in the ground; and 6 = 3 the 
sinkin by the next blow: then rd AN a X 3=2:65=8 

69 : CEST gaan ET dhha cs oid Sug 
the ed stroke. Next substituting : Specimen of the Series 
d +- 6 for d, and 5' for b, the same of the Successive values 
theorem gives 24°8 for the next of d, b, b’. 
sinking, or the next value of b’. | 





And so on continually, by which’ ea i 
means the series of the successive 10 3 2:65 
corresponding values of the letters 13 2-65 | 2-48 
will be as in the margin, the last 15°65| 2-49 | 9-39 
column showing the several suc- | 18°14} 2°32 | 2-19 
cessive sinkings, of the pile by the 20:46] 2°19 | 2-08 
repeated strokes of the ram. Sere > 


Scholium. Thus then it appears that the effect of any ope- 
ration of pile-driving may be determined. It is manifest also 
that the greater a is, or the higher the top of the machine is 
where the ram falls from, above the top of the pile at first, 
the greater will be every stroke of the ram, and consequently 
the fewer the strokes requisite to drive the pile to the requi- 
site depth. But then every stroke will take a longer time, 

Vou. UH. , 63 as 


490 PROMISCUOUS EXERCISES. 


as the ram will be both longer in falling and longer in raising : 
so that it may be a question whether on the whole the busi- 
ness may be effected in the less time by a greater height of 
the machine, or whether there be any limit to the height, so 
as to produce the greatest effect in a given time. 

- To answer this question, let x denote the indeterminate 
height from which any weight w is to fall, z the time of rais- 
‘ing it after a fall, which time is supposed to be as the height 
a to which it is raised, also m the given time of producing a 
proposed effect ; then} ./ x = the time of the weight fall- 
ing ; rang: } o ax “a z = the whole time of one stroke ; 


conseq.. —-—— is the number of strokes made in 
: eye ee AMxw 
the given time mi and hence eae aap = the whole force or 


effeet in the time m. Now this effect or fraction increases 
continually as x increases, because the numerator increases 
faster than the denominator, since the former increases as a, 
while in the latter though the one term z increases as 2, yet 
the other term ,/ x only increases as the root of x. So that, 
on the whole, it appears that the effect, in any given time, in- 
creases more and more as the height is increased, 


PROBLEM 9. 


Fo determine how far a man, who pushes with a fate of 100/b, 
can force a sponge into a piece of ordnance, whose diameter 
as 5 inches, and length ten feet, when the barometer stands at 
30 inches: the vent; or touch-hole, being stopped, and the 
sponge having no windage, that1s, fitting the bore quite close ? 


A column of quicksilver 30 inches high, and 5 in diameter, 
is 5? X30 X-+7854==489-05 inches ; which, at 8-102 oz. each 
inch, weighs 4772-48 oz. or 298:28lb, which is the pressure 
of the atmosphere alone, being equal to the elasticity of the 
air in its natural state ; to this adding the 100lb, gives 398:28lb, 
the whole external pressure. Then, as the spaces which a 
‘quantity of air possesses, under different pressures, are in the 
periprocal ratio of those pressures, it will be, as 398-28 : 
298-28 : : 10 feet or 120 inches : 90 inches nearly, the space 
occupied by the air; theref. 120-—90=30 praches, is the dis- 
tance sought. 
j PROBLEM 10. 


To assign the Cause of the Deflection of Military Proecte 


It having been surmized that in the practice of artillery, the. 
deflexion of the shot in its flight, to the right or left, from the - 
line or direction the gun is laid in, chiefly ¢ arises froii the mo- 

tion 


PROMISCUOUS: EXERCISES. 49} 


tion of the gun during the time the shot is passing out of the 
piece ; it is required to determine what space an 18 pounder 
will recoil or fly back, while the shot.is passing out of the 
gun; supposing its weight to be 4800lb that of the carriage 
2400\b, the quantity of powder 8lb, the length of the cylinder ~ 
' 108 inches, that of the charge 13 inches, and the diameter of 
the bore 5:13 inches ; supposing also that the resistance from 
ihe friction between the platform and carriage is equal to 
3600Ib ? : | 
It is well known that confined gunpowder, when fired, 
immediately changes in a great measure into an elastic air, 
which endeavours to expand in all directions. Now, in the 
question, the action of this fluid is exerted equally on the 
bottom of the bore of the gun and on the ball, during the 
passage of the latter through the eylinder; the two bodies 
therefore move in opposite directions, with velocities which 
are at all times in the inverse ratio of the quantities of matter 
moved. Now let x be the space through which the gun re- 
coils; then, as the charge occupies 13 inches of the barrel, 
and the semidiameter of the barrel is 2°565, the space moved 
through by the ball when it quits the piece, is 108 —- 13 — 
2:565 — x = 92-435 — x: and as the elastic fluid expands 
in both directions, the quantity which advances towards the 
muzzle, is to that which retreats from it, as 92-435 —xtoz: 
: 8x 92:435—z 
eoneed 99-435 “Y° 99-435 
powder which move, the former with the gun, and the latter 
with the ball; besides these, the weight of ball that moves 
forwards being 18lb, and of the weights and resistance back- 
wards 4800 -+- 2400 + 3600 = 10800lb, hence the whole 
8x 
92-435 


92-435—c |. 998298+8e 2403-31 ~ Bx - 
Soap oa? Oh a8 


X 8 are the quantities of the | 


weights moved in the two directions are 10800 -++ and 





Aor Trggape 8! Sgo-496 


the numerators of these only. But when the time and mov- 
ing force are given, or the same, then the spaces are inverse- 
ly as the quantities of matter; therefore x : 92-435: : 
2403-31—8a : 998298+-8x, or by.composition, x : 92:435: : 
2403°31—8x : 1000701°31, and by div. 2 : 1: : 2403-31 = 82 ; 
10826, theref. 10826z = 2403°31—8z, or 108347 = 2403°31, 
and hence «=:2218 inch = 2 ofan inch nearly, or the recoil 
of the gun is less than a quarter of an inch, 
Hence it may be concluded, that so small a recoil, straight 
- backwards, can have no effect in causing the ball to deviate 
from the pointed line of direction : and that it is very probable 
we 


402 PROMISCUOUS EXERCISES. 


we are to seek for the cause of this effect ‘in the ball striking 
or rubbing against the sides of the bore, in its passage through 
it especially near the exit at the muzzle; by which it must 
happen, that if the ball strike against the right side, the ball 
will deviate to the left ; if it strike on the left side, 1t must de- 
viate to the right ; if it strike against the under side) it must 
throw the bal! upwards, and make it to range farther ; but if it 
strike against the upper side it must beat the ball downwards, 
and cause a shorter range :. all which irregularities are found 
to take place, especially in guns that have much windage, or 
which have the balls too small for the bore. 


PROBLEM ‘11. — 


A ball of lead, of 4 inches diameter, is dropped from the top of 
a tower, of 65 yards high, and falls into a erstern full of wa- 
ter at the bottom of the tower, of 20} yards deep : wt 1s requir- 
ed to determine the times of failing, both to the surface and to 
the bottom of the water. 


* The fall in air is 195 feet, and in water 60E eat By the 
common rules of descent, as of 16: 4/ 1957: 1%: 27195 = 
3°49 eenue, the time of descending in air. And as WY 16: 
/195 ::32 : 84/195= 111-71 feet, the velocity at the end 
of that time, or with which the ball enters the water. 


Again, . Re 22 of this vol. art. 2, the aie s== ap Xbyp- 








log of © maa a or rather X hyp. log. of © py ~ (the velocity 


being decreasing and e? greater than a) =F xX com. log. of 


= where n == 11325 the density of lead, n = 1000 that 


ridin lead Sy b= se ,¢ = 111°71 the Pelee. 
(ay Bn 8dn 
ty at entering the water, and v the velocity at any time after- 
wards, also d the diameter of the ball = 4 uinmil and m= 
2-302585 the hyp. log. of 10. 
Hence tet N= 11325, n= 1000, N—n= 10325, a= 


C2 mm 





i 


of water, a = 





4 1 _ 256d(n—n) _ 256... 10325, 

a ee = then log 

iar antes 3n Pegag oe ee 
3 9n' 9000 15 1 

b= = oy _ 9000 = —— = —nearly. Alsoe=111- “71; 


2—a 








m ce? —& 
therefore s $= 602= oR Jog. of ig ox log. - eran 


This 


PROMISCUOUS EXERCISES, 493 


This theorem will give s when v is given, and by reverting 
it will give v in terms of s in the hewn manner. 


= ns, by putting 





Dividing by 5m, gives = = log. of 


1 cate el e? — 
n= —; therefore, the palura) number is 10%, = an ; 
5m’ v2 —a’ 

e2 = 
hente oot aon. oe , and v=,/ (a +o OY al, by sub- 
stituting the numbers above mentioned iy “the letters, gives 
v = 17-134 for the last velocity, when the space s = 603, or 
when the bali arrives at the bottom of the water. 

But now to find the time of passing through the water, 
putting ¢ = any time in motion, and s and v the correspond- 
ing space and velocity the general theorem for variable ae 


gives ¢ = But the above teak value of 's being = x 











+ fea 
hyp. 108: 3 : : tharetore its fluxion 
5 10v% —, conseq. ¢ or — = = 10% , the correct finent of 
cana v We oh 
5 v/a 
which is —— X h toe ring ty the time, 
var ee (> = CoN aa) 


which when v = 17: 134, or s= - 608 gives a 6542 seconds, 
for the time of descent through the water. 


PROBLEM 12. 


Required to dere id what must be the diameter of a water- 
wheel, so as to receive the greatest effect from a stream of water 


of 12 feet fall: ? 


In the case of an undershot wheel 
put the height of the water an = 12. 
feet == a, and the radius sc or cp of 
the wheel==«, the water falling perpen- 
dicularly on the extremity of the radius 
cpatp. Then acor ap=a-—z, and the 
velocity due to this height, or with which 
the water strikes the wheel at p, will be 
as ,/(a —a), and the effect on the wheel being as the velocity 
and as the length of the lever cp, will be denoted by x,/(a—2) 
Or 4f (ax? —x3), which therefore must be a maximum, or its 
square ax? x? amaximum.- In fluxions 2ax7—3a722 = 0; 
and hence x = 2a = 8 feet, the radius. e 
ut 





494 PROMISCUOUS EXERCISES. 


But if the water be considered’ as. 
- conducted so as to strike on the bottom 
_of the wheel, as in the annexed figure, 
it will then strike the wheel with its 
greatest velocity, and there can be no 
limit to the size of the wheel, since the 
greater the radius or lever BC, the 
greater will be the effect.* 

In the case of an overshot wheel, 
a~2x will be the fall of water, ,/(a—2x) 
as the velocity, and x ,/ (a — 2x) or 
a/ (ax? ~ 2%) the effect, then ax?-2x% 
is a maximum, and 2axrz7 — 6227 = 0; 
hence « = 1a = 4 feet is the radius of 
the wheel. 

But all these calculations are te be considered as independ- 
ent of the resistance of the wheel, and of the weight of the 
water in the buckets of it. 





PROBLEM 13. 


What angle must 0 projectile make with the plane of the‘horizon, 
discharged with a given velocity v, so.as to describe in tts flight 
a parabola including the greatest area possible ? 


By the set of theorems (in art. 92 Projectiles) for any 
proposed angle, there can be assigned expressions for the 
horizontal range and the greatest height the projectile rises 
to, that is the base and axis of the parabolic trajectory. Thus, 
putting s and c for the sine and cosine of the angle of eleva- 
tion; then, by the first line of those theorems, the velocity 
being v, the horizontal range Rn Is = j,scv? ; and, by the 4th 
or last line ef theorems, the greatest height 4 is = {4 s?v?. 
But, by the parabola, 2 of the product of the base or range 
and the height is the area, which is now required to be the 
greatest possible. T herefore rR XH = phe? X15? v2 must 
be a Maximum, or, rejecting the constant factors, ‘s8c a maxi- 
mum. But the cosine c, of the angle whose sine’ is, $, 18 

/ (1—s?); therefore s?c=s*,/(1—s?) = Jf (8° ~—s®) is the 
an or its square s§—s*%a HOI In fluxions 
6355 — S579 = = 0= 3 — 4s? ; hence 4s? = 3, or s? = 3, and 
s= 44/3 = -8660254, the sine of 60°, which is the angle of 
elevation to produce - a ehciesd trajectory of the greatest 
area. Patio 





~*: This result 3 is inadmissible. When the investigation is: properly conduebal, 
it is found that the ether} is the same, whatever be the radius of the wheel. 
PROBLEM 


PROMISCUOUS EXERCISES, 495 


PROBLEM 14. : 


Suppose a cannon were discharged ata point A; itis required 
to determine how high in the air the point c must be raised 
above the horizontal line az, so that a person at c letting fall 
a leaden bullet at the moment of the cannon’s explosion, tt may 
arrive at B at the same instant as he hears the report of the 
cannon, but not till 3,th of a secund after the sound arrives 
at B: supposing the velocity of sound to be 1140 feet per se- 
cond, and that the bullet Hod freely without any resistance from 
the air ? 


Let denote the time in which the oMaey 
sound passes toc; then will e— 5 be 
the time in passing to B, and x the time 
also the bullet is falling through cs. 
Then, by uniform motion, 1140z = ac, Rae R 
and 11402—114 = = also by descents : 
of gravity, 17: ©? :: 16: 1622=zc. Then, by right-angled 
triangles, ac? —pc? =aB?, that is 114022? — 16244==1140203 
—224X 11407+114?, henge 224 X11400— 16224 =114?, or 
1015°3%—a24 =5077, the root of which equa. is 7=10-03 se- _ 
conds, or nearly 10 seconds ; conseq. sc=16x2=1610 feet 
nearly, the height required. 


‘PROBLEM 15. 


Required the quantity, in cubic feet, of light earth, necessary to 
form a bank on the side of a canal, which will just support a 
pressure of water 5 feet deep, and 300 feet long. And what 
will the carriage of the earth: cost, at the rate of 1 shilling per 
ton ? 


This question may be considered as 
relating é¢ither to water sustained by a 
solid wall, or by a bank of lose earth. 
In the former case, let asc denote the 
wall, sustaining the pressure of the wa- 
ter behindit. Pat the whole altitude az 
= a, the base sc or thickness at bottom 
= b, any variable depth av = x, and ~ 
the thickness there pe = y. Now the effect ‘which any num- 
ber of particles of the fluid pressing at p have to break the 
wall at B, or to overturn it there, is as the number of particles 
AD or x, and asthe lever sp = a = x: therefore the fluxion 

of the effect of all the forces is (a-—x)tx = axx — 2x, the 

~ fluent of which is Laz? — 1x3, which, when x = a, is 14° for 
the whole effect to break or: overturn the wall at 8 ; and the 
effects 





496 PROMISCUOUS EXERCISES. 


effects of the pressure to break at 8 and p will be as an? and 
ap’, But the strength of the wall at p, to resist the fracture 

there, like the lateral strength of timber, is as the square of. | 
the thickness, px?.. Hence the curve line Arc, bounding 
the back of the wall, so as to be every where equally strong. 
is vif such a nature, that 2? is always proportional to y?, or y 


asx? , and is therefore what is’ called the semicubical parabola. 
Now, to find the area anc, or content of the wall bounded 
by this COnY ER CAINE, the general Hexion pf: all are as yx be- 


comes x22, the fluent of which is act=aen? = zy, that 1s 2 
of the rectangle as XxBc; and is therefore less than the trian- 
gle anc, of the same base and height, in the proportion, of 2 
to 4, or of 4 to 5. | 


But in the case of a bank made of 
earth, it would not stand with that 
concave form of outside, if it were ne- 
cessary, but would dispose itself in a 
straight line ac, forming a triangular 
bank asc. And even if this were not 
the case naturally, it would be preper 
to make it such by art ; because now BR 
neither is the bank to be broken as with thé effect of the 
Jever, or overturned about the pivot or point c, nor does it 
resist the fracture by the effect of a lever, as before ; but, on 
the contrary, every point is attempted to be pushed horizon- 
tally outwards, by the horizontal pressure of the water, and 
it, is resisted by the weight or resistance of the earth at any 
part pe. Here then, by hydrostatics, the pressure of the 
water against atiy point p, is as the depth ap; and, in the 
triangle of earth ape, the resisting quantity in pe is as DE, 
which i is also proportional to ap by similar triangles. So that, 
at every point p in the depth, the pressure of the water and 
the resistance of the soil, by means of this triangular form in- 
crease in the same proportion, and the water and the earth 
will every where mutually balance each other, if at any one 
point, as B, the thickness sc of earth be taken such as to ba- 
lance the pressure of the water at x, and then the straight 
line ac be drawn, to determine the outer shape of the earth, 
All the earth that is afterwards placed against the side ac, for 
a convenient breadth at top for a walking path, &c. will also 
give the whole a sufficient security. 

But now to adapt these principles to the Super calcula- 
tion proposed i in the question; the pressure of water against © 
the point B being Senoted by the side as = 5 feet, and the 

weight 





PROMISCUOUS EXERCISES. 497 


weight of water being to earth as 1000 to 1984, therefore as 
1984 : 1000 :: 5 : 2-52—= Bc, the thickness.of earth which 
will just balance the pressure of the water there ; hence the 
area of the triangle anc = lap x Bc = 21 X 2:52 = 6:3; 
this mult. by the length 300, gives 1890 cubic feet for the 
quantity of earth in the bank ; and this multiplied by 1984 
ounces, the weight of 1 cubic foot, gives, for the weight of it, 
3749760 ounces = 234360lbs = 104°625 tons ; the expense 
ef which, at 1 shilling the ton, is 51. 4s. 71d. 


PROBLEM 16. 


A person standing at the distance of 20 feet from the bottom of a 
wall, which is supposed perfectly smooth and hard, desires to 
know in what direction he must throw an elastic ball against 
at, with a velocity of 80 feet per second, so that, after reflec- 
“tion from the wall, zt may fall at the greatest distance possi- 
ble from the bottom on the horizontal plane, which is 24 feet 
below the hand discharging the ball. 


In the annexed figure let pr ~ r 
be. the wall against which the , 
ball is thrown, from the point 4 
A, in such a direction, that it a: 
shall describe the parabolic curve { 
aE before striking the wall, and =~ “D | 
afterwards be so reflected as to describe the curve ep. Now 
if Es be the tangent at the point £, to the curve ar describ- 
ed before the reflection, and rer the tangent at the same 
point to the curve which the ball will describe after re- 
flection, then will the angle rer be = cus ; and if the curve 
re be produced, so as to have ar for its tangent, it will meet 
ac produced in B, making pc = ao, and the curve ag will be 
similar and equal to the portion Be of the parabola Ber, but 
turned the contrary way. Conceiving either the two curves 
ar and ep, or the continued curve Ber, to be described by a 
projectile in its motion, it is manifest that, whether the great- 
er portion of the curve be described before or after the ball 
reaches the wall pr, will depend on its initial velocity, and on 
the distance ae or gc, and on the angle of projection. The 
problem then is now reduced to this, viz. ‘To find the angle 
at which a ball shall be projected from g£, with a given impe- 
tus, so that the distance pp, at which it falls, from the given 
point » on the plane pp, parallel to the horizon, shall be a 
maximum. , 

Vor. Uf. 64 . Now - 






498 PROMISCUOUS EXERCISES. 


Now this problem maybe pF 
_ constructed in the following 
manner ; From any point & 

in the horizontal line pc, let ; 
fall the indefinite perp. zc, 8B : 
on which set off es =the A p 
impetus corresponding to the G 

given velocity, and gi = 2} | 

the distance of the horizontal plane below the point of pro- 
jection ; also, through 1 draw apr parallel to nc. From the 
point B set cite = sE -+ EI, and bisect the angle ezp by the 
line BH: then will Ba be the required direction of the ball, 
and 1p the maximum range on the plane ap. 

For, since the ball moves from the point 8 with the veloci- 
ty acquired by falling through sp, it is manifest, from p. 136 
this vol. that nc is the directrix of the parabola described by 
the ball. And since both gp and P are points in the curve, 
each of them must, from the nature of the parabola, be as far 
from the forces as it is from the directrix ; therefore 8 and P 
will be the greatest distance from each other when the focus 
F is directly between them, that is, when BP = nE-+ cr. And 
when sp is a maximum, since Br is constant, it is obvious that 
iP is a maximum toe. Also, the angle rsa being = egn, the 
line su is a tangent to the parabola at the point 8, and consé- 
quently it is the direction necessary to give the range 1p. 

Cor. 1. When 8 coincides with 1, rp will be = sp=sn--Et 
=2e1, and the angle een will be 45°: as is also manifest from 
the common modes of investigation. 

Cor. 2. When the impetus corresponding to the initial ve- 
locity of the ball is very great compared with ac or gc (fig. 1,) 
then the part az of the curve will very nearly coincide with 
its tangent, and the direction and velocity at a may be ac- 
counted the same as those at e without any sensible error. In 
this case too the impetus Be (fig. 2) will be very great com- 
pared with si, and consequently, 8 and 1 nearly coinciding, 
the angle cpu will differ but little from 45°. 

Calcul. From the foregoing construction the calculation 
willbe very easy. Thus the first velocity being 80 feet =U, 


U4 80. 
then (art. 92 Projectiles) — Fs oe ee p=88: 48186 = pe the 





1 


impetus ; hence c1 = ep = 101+98186, and sp = Be ++ et = 
201°46372. Now, in the right-angled triangle pir, the sides. 
gr and Bp are known, hence re = 201°4482, and the angle 
1BP=89° 17’ 20”: half the suppl. of this angle i is 45° 21’20" 
=epeH. And, in fig. 1,1P—1m =201-4482— 10=191-44829= 


pp; the distance the ball falls from the wall after reflection: 
PROBLEM 


PROMISCUOUS EXERCISES. 499 
PROBLEM 17. 


From what height above the given point a must an elastic ball be 
suffered to descend freely by gravity, so that after striking the 
hard plane at B, it may be reflected back again to the point a, 
in the least time possible from the instant of dropping it ? 


Let c be the point required ; and put ac = z, and Cc 
ap == a; then} ,/ cs =1,/(a-+2) is the time in cp, 
and },/ca==1 y x is the time inca; thereforely/ [A 
(a+2)—14/x is the time down as, or the time of ris- 
ing from B to 4 again: hence the whole time of fall- 
ing through cz and returning to a,is3,/(a+2) —2 
4/«, which must be a min. or 2 ./ (a--2)— ,/z a mi- 


nimum, in fluxions = ne O, and hence ih 
1 ren ees ax=la 
ee J(ate) fn ue 
that is, ac=Lag, 

PROBLEM 18. 


Given the height of an inclined plane ; required its length so 
that a given power acting on a given weight, in a direction pa- 
rallel to the plane, may draw wt up in the least time possible ? 


Let a.denote the height of the plane, ~« its length. p the 
power, and w the weight. Now the tendency down the plane 
7 aw 


e aw aw ish : Ee 
is = —, hence p—— = 'the motive force, and -———— 
x eS _ptw 


cio the accelerating force f ; hence, by the theorems for 
constant forces (See Introduc. to Prac. Ex. on Forces)##=— = 
2 Hie BY 
dard me must be a minimum, or “amin. ; in fluxions, 
(px— aw )g 3 pt —aw peer 
2(pxr — aw)xx — px*x = 0, or px = 2aw, and hence p: 
w:: 2a: x :: double the height of the plane to its length. 








PROBLEM 19. 


A cylinder of oak is depressed in water till its top is just level 
with the surface, and then is suffered to ascend ; it 1s required 
to determine the greatest altitude to which it will rise, and the 
tume of its ascent. | 
Let a = the length, and b the area or base of the cylinder, 

m the specific gravity of oak, that of water being 1, also x 

any variable height through which the cylinder has ascended. 

Then, a—x being the part still immersed in the water, 


(as) 


500 PROMISCUOUS EXERCISES. 


(a~x) XbX1=(¢— Pe b is the force of the water upwards 
to raise the cylinder; anda x 6 X m = abm is the weight 
of the cylinder opposing its ascent ; therefore the efficacious 
force io raise the cylinder is (a — 2) b—abm; and, thet mass 
being abm, the accelerating force is 
(a—x)b—abm  a—x—-am _ on—x __ 
abm ae am =fs 
putting n = 1 — m the se deh between the specific gra- 
vities of water and oak. 
Now if v denote the velocity of ascent at the same time 
when x space is ascended, then by the theorems es variable 


32 
forces, vy = 32fz = es Xx (anx—xx), therefore 








9 naw 
v2 = X (Lanz -2?), and v = 8 / act but when 
the cylinder has acquired its greatest ascent, v and v? = 0, 
therefore Yanx — «?.= 0, and hence « = 2an the Pant of the 
cylinder that rises out of the water, beer = ‘15a or 33, of its 
length. 
bs To find when the velocity i is the greatest, the factor 2anx 
—x2 in the velocity must be a.max. then 2anz—2rx = 0, 
and x = an, being the height above the water when the ve- 
locity is the greatest, and which it appears is just equal to the 
half of 2an above found for the greatest rise, when the up- 
ward motion ceases, and the cylinder descends again to the 
same depth as at first, after which it again returns ascending 
as before ; and so on, continually playing up and down to the 
same highest and lowest points, like the vibrations of a pen- 
dulum, the motion ceasing in both cases in a similar manner at 
the extreme pomts, then returning, it gradually accelerates 
till arriving at the middle point, where it is the greatest, then 
eradually retarding all the way to the next extremity of the 
vibration, thus making all the vibrations in equal times, to the 
same extent between the highest and lowest points, except 
that, by the smail tenacity and friction &c. of the water against 
the sides of the cylinder, it will be gradually and slowly re- 
tarded in its motion, and the extent of the vibrations decrease 
till at length the cylinder, like the pendulum, come to rest in 
the middle point of its vibrations, where it naturally floats in 
its quiescent state, with the part na of its length above the 
water. 
‘The quantity of the greatest velocity will be found, by 


substituting na for a, in “the general value of the velocity 
Zane — x? 


t 3 Vv Sam | 5 when it becomes Sn —=4 Jf G very nearly, 


the 





PROMISCUOUS EXERCISES. 50| 


the value of m being “O25, and consequently that of n=1— 
m = ‘075. 


To find the time ¢ answering to_ any space a Hard 





ig x ma x 
(= SS andb the 
Dg ., Inaz—2? Vv 3 af (2nax—x?) y 
| Ima 


13th form the fluent is ¢ = 1,/2ma X a, Bete A denotes the | 


. ° | 
circular arc to radius 3 and versed sine =. Now at the mid- | 
| nr 


~ 


na 


’ ne 
the radius, and a is the quadrantal arc=1-5708 ; then the flu. 


becomes 4 Lf 2ma X 15708 = +17 fa X 1 5708 = 28 T4/ a 
for the time of a semivibration ; hence the time of each whole 
vibration 1s *534 ,/ a = ;%;o/a, which time therefore depends 
on the length of the cylinder a. To make this time = 1 
second, a must be =(1p)2 very nearly=31 feet or 42 inches. 
That is, the oaken cylinder of 42 inches length makes its 
vertical vibrations each in 1 second of time, or is isochronous 
with a common pendulum of 391 inches long, the extent of 
each vibration of the former being 63, 7 inches. 


dle of a vibration x is = na, and then the vers. =~ a 


PROBLEM 20, 


Required to determine the quantity of matter in a sphere, the den: 
sity varying as the nth power of the distance from the centre!’ 


Let r denote the radius of the sphere, d the density at 
its surface, a = 3°1416 the area of a circle whose radius is 1, 
and x any distance from ‘the ‘centre.’ Then 4¢z2 will be the 
surface of a sphere whose radius is 2 which may be‘consi- 
dered by expansion as generating the magnitude of the solid 
therefore 4ax? willbe the fluxion of the magnitade; but 





: di in : x f r 
as 72a? s3d ve the. density at the distance 9, LReERT Gre 
jf 

dx” ‘4ad: 

Aa? xX es heer =2 = the fluxion of the mus the fluent 
rt r 

4 #3 

of which -— ae 





> when r=r, ie -y the anit aE the 
matter in “the a eae sphere. 


Corol. 1. The magnitude of a my hetion radius is 7 
bemg gars, which call m; then the ae or solid content will 





be ara* m, and the mean density is 


aa 
Corol, 


It having been computed, from actual experi- 
| ments, 





502 PROMISCUOUS EXERCISES. 


ments, that the medium density of the whole mass of the 
earth is about twice the density d at the surface, we can 
now determine what is the exponent of the decreasing ratio 
of the density from the centre to the circumference, sup- 
| posing it to decrease by a regular law, viz. as 7” ; for then it 


will be oe te 


° pest: Ti 
the law of decrease is as” 2, or as 





, and hence n=—3. So that, in this case 


1 5 ee 
—, that is inversely as 
ri 


the 2ths power of the radius. 
PROBLEM 21. 


Required to determine where a body moving down the convex 
side of a cyelord, will fly off and quit the curve. 


- Let aves represent the cy- 
cloid, the properties of which 
may be seen at arts. 146 and 
147 this vol. and voc its gene- 
rating semicircle. Let £ be the 
point where the motion com- 
mences whence it moves along the curve, its velocity increas- 
ing both on the curve, and also in the horizontal direction pr, 
till 1t come to such a point, F suppose, that the velocity in 
the latter direction is become a constant quantity, then that 
will be the point where it will quit the cycloid, and after- 
wards describe a parabola rc, because the horizontal velocity 
in the latter curve is always the same constant quantity, (by 
ari, 76 Projectiles.) - | 
Put the diameter vc = d, vu =a, vi=z ; then vp=,/ dz, 
and 1p = 4/ (dx — x2). Now the velocity in the curve at p 
in descending down er, being the same as by falling through 
HI or g~a, by art. 139, will be = 8 ,/ (cx—a) ;_ but this ve- 
locity inthe curve at F, is to the horizontal velocity there, 
as vp to tp, because vp is parallel to the curve or to the 
tangent at r, that is, / dx: ./ (dx —a?):: 84 (e@—a): 


i bible seth A eate) , which is the horizontal velocity at r, 





where the body is supposed to ‘have that velocity a constant 
quantity ; therefore also ,/ (x — a) X ,/f (d — =), as well as 
(c—a) X (d—x) = ax + dx — ad — x? 1s a constant quan- 
tity, and also az -+- dz — «2: but the fluxion of a constant 
quantity is equal to nothing, that is az + dx — 2rz =O = 
a-—+ d — 2x, and hence zs = 1a -+-1d = vi, the arithmetical 
mean between vu and ve. : ; 

If the motion should commence at v, then x or vi would be 
==1d, and 1 would be the centre of the semicircle. 


PROMISCUOUS EXERCISES. £03 


PROBLEM 22. 


Hf a body begin to move from a, with a given velocity, along 
the quadrant of a circle ap ; it is required to show at what 
point it will fly of from the curve. : 


Let p denote the point where the 
body quits the circle app, and then de- | 
scribes the parabola se. Draw the or- 
dinate pr, and let ca be the height 
producing the velocity at a. Put ca= 
a, AC Or cb=r, ar=z ; then the velo- 
city in the curve at p will be the same 
as that acquired by falling through cr 
or a+, which i is, as before, B4/(a--2) ; ; 
but the velocity in the curve is to the horizontal velocity as 
pn to mn or as cp to cr by similar triangles, that is, as 7 


=: Bf (ea): 8 (xa) X——, which is to be 


a constant quantity where the body leaves the circle, there- 
fore also (r — x) 4/ (x +) and (r — x)? X (a-+a) a con- 
stant quantity ; the fluxion of which made to yaniah gives. x 


_ tm 2a a 
Heng = 








Hence, if a = 0, or the body only commence motion at a, 
then z =r, or ar =1ac when it quits the.circle atv. But 
if @ or Ga were =1r or 4 x Ac, then r — 2a ==0, and the body 
would instantly quit the ‘circle at the vertex a, and describe 
a parabola circumscribing it, and having the same vertex a. 


PROBLEM 23. 


To determine the position of a bar or beam az, being supported 
in equilibrio by two cords ac, Bc, having thetr two ends fixed 
an the beam, at a and B. 


By art. 210 Statics, the position 
will be such, that its centre of gra- 
vity c will be in the perpendicular 
or plumb line ce, 

Corol. 1. Draw ep parallel to the 
cord ac. Then the triangle cep, 
having its three sides in the direc- 
tions of, or parallel to, the three 
_ forces, viz. the weight of the beam, 
and the tensions of the two cords ac, Bc, these three forces 
will be proportional to the three sides ca, cp, cv, usenet 

vy 





504 PROMISCUOUS EXERCISES. 


by art. 44; that is, ca is as the weight of the beam, ep as the 
tension or force of ac, and cp as the tension or force of Bc. 

Corol. 2. If two planes Ear, ner, perpendicular to the two 
cords, be substituted instead of these, the beam will be still 
supported by the two planes, just the same as before by the 
cords, because the action of the p!anes is in the direction per- 
pendicular to their surface ; and the pressure on the planes 
will be just equal to the tension or force of the respective 
cords. So that it is the very same thing, whether the body 
is sustained by the two cords ac, Bc, or by the two planes er, 
wi; the directions and quantities of the forces acting at a and 
B being the same in both cases.—Also, if the body be made 
to vibrate about the point c, the points a, » will describe cir- 
cular ares coinciding with the touching planes at a, B ; and 
_moving the body up and down the planes, will be just the same 
thing as making it vibrate by the cords ; consequently the 
body can only rest, in either ease, when the centre of gravity 
is in the perpendicular ce. . 


PROBLEM 24. 


To determine the position of the beam as, hanging by one cord 
acs, having its ends fastened at a and B, and sliding freely 
over a tack or pulley fixed at c. 


e being the centre of gravity of the beath, ce will be per- 
pendicular to the horizon, as in the last problem. Now as 
the cord acs moves freely about the point 
e, the tension of the cord is the same in 
every part, or the same both in ac and Bc. 
Draw «ep parallel to ac: then the sides of 
the triangle cep are proportional-to the 
three forces, the weight and the tensions. 
of the string ; that is, cp and p¢ are as the 
forces or tensions in cp and ca. But these 
tensions are equal ; therefore co=pe, and conseq. the oppo- 
site angles nce and pec are also equal; but the angle pec is 





= the alternate angle acc; theref. the angle acc=sce ; and | 


hence the line ce bisects the vertical angle: ACB, and conseq. 
AC: CB: 2 AG: GB. 


PROBLEM 2. 


To Paiihine the position of the Sarin AB, moveable about the 
end B, and sustained by a given weight g, hanging by a cord 
ACS, going over a pulley at c, and fixed to the caieee end a. 

Let 


PROMISCUOUS EXERCISES. 505 


Let w= the weight of the beam, G 
and « denote the place of its cen- | 


tre of gravity. Produce the di- ~~ @ f 

rection of the cord ca to meet the i, 

horizontal line BE in D; also let 

fall az perp. tose: thenanisthe DE 

direction of the weight of the beam, 5 

and pa the direction of the weight g, the former acting at c 

by the lever se, and the latter at a, by the lever sa ; theref. 

the intensity of the former is w X Be and that of the latter g 

xX BA; but these are also proportional to the sines of their 

angles of direction with as, that is, of the angles par, and BAD ; 

therefore the whole intensity of the former is w X Ba X sip. 

BaAE, and of the latter itis gXBaX sin Bap. But, since these 

two forces balance each other, they are sh ae viz. wX Ba X 

Sin. BAE=g x BA X sin. Bap, and therefore w: g:: Ba X sin. 

BAD ; BG X SiN. BAZ, Or W X BG: § XBA:: Sin. BAD: SiN. BAE. 

PROBLEM 26. 

To determine the position of the beam ap sustained by the given 
weights m, n, by means of the cords acm, Bon, going over 
the fixed pulleys c, D 
Let c be the place of the centre 

of gravity of the beam. Now the 

effect of the weight m, is as m, - 
and as the lever ac, and as the @ ‘ 
sine of the angle of direction a; 72 a te 
and the effect of the weight n, is ASSSS=/p 

as n, and as the lever se and as GC. 

the sine of the angle of direction 8; but these two effects 

are equal, because they balance each other; that is,m X ac 

X sin. A =n X Ba X sin. B; theref.m X ac: 2X Ba : 

sin. B : sin. A. 








PROBLEM 27. 
To determine the position of the two posts av and BE, support- 
ing the beam as, so that the beam may rest in equilibrio. 


Through the centre of gravity 
c of the beam, draw ce perp. to 
the horizon ;. from any point c 
in which draw cap, cBe through 
the extremities of the beam; then 
ap and BE will be the positions 
of the two posts or props re- 
quired, so as AB may be sustain- 
-ed in equilibrio ; because the 
three forces sustaining any body 
in such a state, must be all directed to the same point c. 
Vou. If. 65 Boro. 





506 PROMISCUOUS EXERCISES. 


Corol. If cr be drawn parallel to cp; then the quantities 
of the three forces balancing the beam, will be proportional 
to the three sides of the triangle cor, viz. ce as the weight 
of the beam, cr as the thrust or pressure in Bg, and FG as the 
thrust or pressure in aD. 

Scholium. ‘The equilibrium may be equally maintained by 
the two posts or props ap, BE, as by the two cords ac, Bc, 
or by two.planes.at a and B perp. to those cords.—lIt does 
not always happen that the centre of gravity is at the lowest 
place to which it can get, to make an equilibrium ; for here 
when the beam as is supported by the posts pa, Es, the cen- 
tre of gravity is at the highest it can get; and being in that 
position, it is not disposed to move one way more than another, 
and therefore is as truly in equilibrio, as if the centre was at 
the lowest point. It is true this is only a tottering equili- 
brium, and. any the least force will destroy it ;, and then, if 
the beam and posts be moveable about the angles a, 8, pD, E, 
which is all along supposed, the beam will descend till it is 
below the points p, £, and gain such a position as is described 
in prob. 26, supposing the cords fixed at c and p, in the fig. 
to that prob. and then c will be at the lowest point, coming 
there to an equilibrium again, In planes, the centre of gra- 
vity c may be either at its highest or lowest point. And 
there are cases, when that centre is neither at its highest nor 
lowest point, as may happen in the case of bape 24. 


PROBLEM 28. 


Supposing the beam av hanging by a pin at B, and laying on the 
. wall ac; itis required to determine the forces Bi img hit at 
the points A and g, and their directions. 


Draw ap perp. to as, and through 
a, the centre of gravity ‘of the beam, 
draw cp perp. to the horizon; and 
join: Bp. Then the weight of the 
beam, and the two forces or pres- 
sures at a and B, will be in tke di- 
rections of the three sides of the 
triangle anc; or in the directions 
of, and proportional to, the three 
sides - of the triangle GDH, having = 
drawn cH parallel to ep; yiz. the Weight of the beam as GD, 
the pressure at a as uD; and the pressure & as GH, and in 
these directions. 

For, the action of the beam is in the direction cp; and the 
action of the wall at 4, is in the perp. ap ; conseq. the stress 

‘on 





PROMISCUOUS EXERCISES. 507. 


on the pin at 8 must be in the direction sn, because all the 
three forces sustaining a.body in equilibrio, must tend to the 
same point, as D. 

Corol. 1. If the beam were supported by a pin at a, and 
laid upon the wall at 8; the like construction must be made 
at B, as has been done at a, and then the forces and their di- 
rections will be obtained. 

Corol. 2. It is all the same thing, whether the beam is sus- 
tained by the pin p and the wall ac, or by two cords Br, ar, 
acting in the directions Bp, pa, and with the forces ua, up. 


™~ 


PROBLEM 29, 


To determine the Quantities and Directions of the Forces exerted 
by a heavy beam ap, at tts two Extremitres and its Centre of 
Gravity, bearing against a perp. wall at tts upper end B. 


From gs draw sc perp. to the face of 
the wall se, which will be the direc- 
tion of the force atz; also through c, 
the centre of gravity, draw cep perp. 
to the horizontal line ag, then cn is the 
direction of the weight of the beam ; 
and because these two forces meet in 
the point c, the third force or push a, 
must be in ca, directly from c; se that the three forces are in 
the directions cp, Bc, ca, or in the directions cp, pa, ca; and, 
these last three. forming a triangle, the three forces are not 
only in those directions, but are also proportional to these 
three lines; viz. the weight in or on the beam, as the line cp; 
the push against the wall at B, as the horizontal line ap; and 
the thrust at the bottom, as the line ac. 

Some of the foregoing problems will be found useful in dif- 
ferent cases of carpentry, especially in adapting the framing 
of the roofs of buildings, so as to be nearest in equilibrio in 
all their parts. And the last problem, in particular, will be 
very useful in determining the push or thrust of any arch 
against its piers or abutments, and thence to assign their 
thickness necessary to resist that push. The following prob- 
lem will also be of great use in adjusting the form of a man- 
sard roof, or of an arch, and the thickness of every part, so 
as to be truly balanced in a state of just equilibriam. 





PROBLEM 30. 


Let there be any number of lines, or bars, or beams, an, BC, cp, 
DE, &c. ail in the same vertical plane, connected together and 


freely moveable about the joints or angles a, B, C, D, E, $C. 
and 


508 PROMISCUOUS EXERCISES. 


and kept in equilibrio by their own weights, or by weights only 
laid on the angles: Its required to assign the proportion of 
those weights ; as also the force or push in the direction of the 
said lines; and the horizontal thrust at every angle. 


Through any 
point, as p, draw 
a vertical line 
apug, &c..; to 
which, from any 
point, as c, draw 
lines in the di- 
rection of, or pa- 
rallel to, the giv- G 
en lines or beams, viz. ca parallel to az, and cb parallel to sc, 
and ce to pg, and ¢f to er, and cg to Fa, &c. ; also cn parallel 
to the horizon, or perpendicular to the vertical line ang, in 
which also all these parallels terminate. 

Then will all those lines be exactly proportional to the 
forces acting or exerted in the directions to which they are 
parallel, and of all the three kinds, viz. vertical, horizontal, 
and oblique. That is, the oblique forces or thrusts in direc- 





tion of the bars - - - - - = AB, BC, CD, DE, EF, FG, 
are proportional to their parallels ca, cb, cD, ce, cf, cg; 
and the vertical weights on the angles B, c, pb, ©, F, &c. 
are as the parts of the vertical - - ab, bn, ne, ef, fg. 


and the weight of the whole frame = aBcpEFc, 

is proportional to the sum of all the verticals, or to ag ; also 
the horizontal thrust at every angle, is every where the same 
constant quantity, and is expressed by the constant horizon- 
tal line cu. 

Demonstration. All these proportions of the forces derive 
and follow immediately from the general well-known property 
in Staties, that when any forces balance and keep each other 
in equilibrio, they are respectively in proportion as the lines 
drawn paraliel to their directions, and terminating each other. 

Thus, the point or angle 8 is kept in equilibrio by three 
forces, viz. the weight laid and acting vertically downward 
on that point, and by the two oblique forces or thrusts of the 
two beams aB, cx, and in these directions. But ca is parallel 
to aB, and cb, to sc, and ab, to the vertical weight; these 
three forces are therefore proportional to the three lines ab, 
60, chs 

In Jike manner, the angle c is kept in its position by the 
weight laid and acting vertically on it, and by the two oblique 
forces or thrust in the direction of the bars Bc, cp; conse- 

quently 


PROMISCUOUS EXERCISES. 509 


quently these three forces are proportional to the three lines 
bp, cb, cp, which are parallel to them. 

Also, the three forces keeping the point p in its position, 
are proportional to their three parallel lines pe, cp, ce. And 
the three forces balancing the angle x, are proportional to 
their three parallel lines ef, ce, cf. And the three forces 
balancing the angle Fr, are proportional to their three parallel 
lines fg, cf, cg. And so on continually, the oblique forces 
or thrusts in the directions of the bars or beams, being always 
proportional to the parts of the lines parallel to them, inter- 
cepted by the common vertical line ; while the vertical forces 
or weights, acting or laid on the angles, are proportional to 
the parts of this vertical line intercepted by the two lines pa- 
rallel to the lines of the corresponding angles. 

Again, with regard to the horizontal force or thrust : since 
the line pc represents, or is proportional to the force in the 
direction pc, arising from the weight or pressure on the angle 
pd; and since the oblique force pc is equivalent to, and re- 
solves into, the two pu, Hc, and in those directions, by the re- 
solution of forces, viz. the vertical force DH, and the horizon- 
tal force uc; it follows, that the horizontal iidde or thrust at 
the angle p, is proportional to the line cH; and the part of 
the vertical force or weight on the angle p, which produces 
the oblique force pc, is proportional to the part of the verti- 
cal line pH. 

In like manner, the oblique force cb, acting at c, in the di- 
rection cB, resolves into the two bu, Hc; therefore the hori- 
zontal force or thrust at the angle c, is expressed by the line 
cu, the very same as it was before for the angle p; and the 
vertical pressure at c, arising from the weights on both pand 
c, is denoted by the vertical line bx. 

Also, the oblique force ac, acting at the angle sz, in the di- 
rection BA, resolves into the two au, uc; therefore again the 
horizontal thrust at the angle B, is represented by the line cn, 
the very same as it was at the points c and p ; and the verti- 
cal pressure at B, arising from the weights on zg, c, and Ds is 
expressed by the part of the vertical line an. » 

Thus also, the oblique force ce, in direction px, resolves 
into the two cu, ne, being the same horizontal force with the 
vertical He; and the oblique force cf, in direction er, resolves 
into the two cu, nf; and the oblique force cg, in direction 
FG, resolves into the two cu, Hg ; and the oblique force cg, 
in direction FG, resolves into the two cu, HE 5 and so on con- 
tinually, the horizontal force at every point being expressed 
by the same constant line cn; and the vertical pressures on 
the angles by the parts of the verticals, viz. aH the whole 

vertical 


510 PROMISCUOUS EXERCISES. 


vertical pressure at 8, from the weights on the angles 8, ¢, p ; 
and 6H the whole pressure on c from the weights onc andp ; 
and px the part of the weight on p causing the oblique force 
pe; and ue the other part of the weight on p causing the ob- 
lique pressure pE; and uf the whole vertical pressure at £ 
from the weights on p and ©; and ug the whole vertical pres- 
sure on F arising from the weights laid on p, E, and Fr. And 
go on. 

So that, eon the whole, au denotes the deka weight on the 
points from p to a; and ug the whole weight on the points 
from p to ec ; and ag the whole weight on all points on both 
sides ; while ab, bn, ve, ef, fg express the several particular 
weights, laid on the angles B, c, n, £. F. 

Also, the horizontal thrust is every where the same con- 
stant quantity, and is denoted by the line cu. 

Lastly, the several oblique forces or thrusts, in the direc- 
tions AB, BG, CD, DE, EF, FG, are expressed by, or are propor- 
tional to, their corresponding parallel lines, ca, cb, cp, ce, 
cf, eg. 

Dae 1. %Jtis obvious, and remarkable, that the lengths 

of the bars az, Bc, &c. do not effect or alter the proportions 

of any of these loads or thrusts ; since all the lines ca, cb, ab, 
&c. remain the same, whatever be the lengths of aB, Bc, &c. 

The positions of the bars, and the weights on the angles de- 

pending mutually on each other, as well as the horizontal and 

oblique thrusts. Thus, if there be given the position of pe, 

and the weights or loads laid on the angles p, c, B ; set these 

en the vertical, pu, pb, ba, then cb, ca give the directions or 

positions of cs, Ba, as well-as the quantity or Epes cH of 
the constant horizontal thrust. | 

Corol. 2. If cu be made radius; then it is evident’ that 
ua is the tangent, and ca the secant of the elevation of ca or 
as above the horizon ; also Hé 1s the tangent and cb the secant 
of the elevation of cb or cB; also np and cp the tangent and 
secant of the elevation of cp ; also He and ce the tangent and 
secant of the elevation of ce or pe; also nf and cf the tan- 
gent and secant of the elevation of er; and so on; also the 
parts of the vertical ab, bp, ef, fg, denoting the weights laid 
on the several angles, are the differences of the said tangents 
ef elevations. Hence then in general, 

Ast..The oblique thrusts, in the directions of the bars, are 
to. one another, directly in proportion as the secants of their 
angles of elevation above the horizontal directions ; or, which 
is the same thing, reciprocally proportional to the cosines of 
the same elevations, or reciprocally proportional to the sines 
of the vertical angles, a, b, p, e, f, g, &c. made by th : 
e verti- 


PROMISCUOUS EXERCISES, 514 


eal line with the several directions of the bars ; because the 
secants of any angles are always reciprocally in proportion ag 
their cosines. | 

2. The weight or load laid on each angle, is directly pro- 
portional to the difference between the tangents of the eleva- 
tions above the horizon, of the two lines which form the angle. 

3. The horizontal thrust at every angle, is the same con- 
stant quantity, and:-has the same proportion to the weight on 
the top of the uppermost bar, as radius has to the tangent of 
the elevation of that bar. Or, as the whole vertical ag, is to 
the line cu, so is the weight of the whole assemblage of bars, 
to the horizontal thrust. Other properties also, concerning 
the weights and the thrusts, might be pointed out, but they 
are. less simple and elegant than the above, and are therefore 
omitted; the following only excepted, which are inserted 
here on account of their usefulness. | 

Corol. 3. It may hence be deduced also, that the weight 
or pressure laid on any angle, is directly proportional to the 
continual product of the sine of that angle and of the secants 
of the elevations of the bars or lines which form it. Thus, 
in the triangle bcp, in which the side bp, is proportional to 
the weight laid on the angle c, because the sides of any tri- 
angle are to one another as the sines of their opposite angles, 
therefore as sin. p.: cb :: sin. bcp : bp; that is, bp is as 


sin. bcp 


sm cae X cb; but the sine of angle p is the cosine of the 


elevation pcw, and the cosine of any angle is reciprocally 
proportional to the secant, therefore bp is as sin. bcp X sec: 
pcH X ch; and cb being as the secant of the angle bcu of 
the elevation of bc or sc above the horizon, therefore bp is 
as sin. bcp X sec. bcn x sec. Den; and the sine of bcp 
being the same as the sine of its supplement scp; therefore 
the weight on the angle c, which is as bp. is as the sin. scp 
X sec. pcH X sec. bcu, that is, as the continual product of 
the sine of that angle, and the secants of the elevations. of its 

two sides above the horizon. Pinte. 
Corol. 4. Further, it easily appears also, that the same 
weight on any angle c, 1s directly proportional to the sine of 
that angle scp, and inversely proportional to the sines of 
the two parts ecr, pep, into which the same angle is divided 
by the vertical line cp. For the secants of angles are reci- 
procally proportional to their cosines or sines of their com- 
plements; but scp = cou, is the complement of the eleva- 
- tion bcu, and per is the complement of the elevation pcu ; 
therefore the secant of bc X secant of ncH is reciprocally 
as the sin, scp X'sin, por’; ‘also. the sine of bcp is = the sine 
of 





‘512 PROMISCUOUS: EXERCISES. 


of its supplement scp ; consequently the weight on the angle 
c, which is proportional to sin. bcp X sec. bcH X sec. DeH, 


(a abe when the whole 








is also proportional to ————— 
sin. BCP X sin. DCP 


frame or series of angles is balanced, or kept in equilibrio, by 
the weights on the angles ; the same as in the preceding pro- 
position. 

Scholium. The foregoing proposition is very fruitful in 
its practical consequences, and contains the whole theory of 
arches, which may be deduced from the premises by suppos- 
ing the constituting bars to become very short, like arch 
stones, so as to form the curve of an arch. It appears too, 
that the horizontal thrust, which is constant or uniformly the 
same throughout, is a proper measuring unit, by means of 
which to estimate the other thrusts and pressures, as they 
are all determinable from it and the given positions ; and the 
value of it, as appears above, may be easily computed from 
the uppermost or vertical part alone, or from the whole as- 
semblage together, or from any part of the whole, counted 
from the top downwards. 

The solution of the foregoing proposition depends on this 
consideration, viz. that an assemblage of bars or beams, being 
eonnected together by joints at their extremities, and freely 
moveable about them, may be placed in such a vertical posi- 
tion, as to be exactly balanced or kept in equilibrio, by their 
mutual thrusts and pressures at the joints; and that the effect 
will be the same if the bars themselves be considered as with- 
out weight, and the angles be pressed down by laying on 
them weights, which shall be equal to the vertical pressures 
at the same angles, produced by the bars in the case when 
they are considered as endued with their own natural weights. 
And as we have found that the bars may be of any length, 
without affecting the general properties and proportions of 
the thrusts and pressures, therefore by supposing them to 
become short, like arch stones, it is plain that we shall then 
have the same principles and properties accommodated to a 
real arch of equilibration, or one that supports itself in a per- 
fect balance. It may be further observed that the conclu- 
sions here derived, in this proposition and its corollaries, 
exactly agree with those derived in a very different way, in 
my principles of bridges, viz. in propositions 1 and 2, and 
their erates: 


PROBLEM 31. 


If the whole figure in the last problem be inverted, or turned 
round the horizontal line ac as an axis, tll at be completely 
reversed 


PROMISCUOUS EXERCISES. 513 


reversed, or in the same vertical plane below the first position, 
each angle p, d, Sc. being in the same plumb line; and if 
weights i,k, 1, m,n, which are respectively equal to the weights 
laid on the angles, B, c, D. E, ¥, of the first figure, be now sus- 
pended by threads from the corresponding angles b, c, d, e, f, 
of the lower figure; it is required to show that those weights 
keep this figure in exact equilibrio, the same as the former and 
all the tensions or forces in the latter case, whether vertical or 
horizontal or oblique, will be exactly equal to the correspond- 
ing forces.of weight or pressure or thrust in the like directions 


of the first figure. 





k Ce I. ert 

This necessarily happens, from the equality of the weights, 
and the similarity of the positions, and actions of the whole in 
both cases. Thus, from the equality of the corresponding 
weights, at the like angles, the ratios of the weights, ad, bd, 
dh, he, &c. in the lower figure are the very same as those ab, 
bp, pu, He, Xc. in the upper figure ; and from the equality of 
the constant horizontal forces cu, ch, and the similarity of the — 
positions, the corresponding vertical lines, denoting the weights, _ 
are equal, namely, ab=ab, bp=-bd, pu =-dh. &c, The same. 
‘may be said of the oblique lines also, ca, cé, &c. which being 
parallel to the beams ab, bc, &c. will denote the tensions of 
these in the direction of their length, the same as the oblique 
thrusts or pushes inthe upper figures. Thus. all the corres- 
ponding weights and actions and positions, ia the two situa- 
tions, being exactly equal and similar, changing only drawing 
.and tension for pushing and thrusting, the balance and equili- 
brium of the upper figure is still preserved the same in the 
hanging festoon or lower one. 

Scholium. The same figure, it is evident, will also arise, if 
the same weights, 7, k, 1, m, n, be suspended at like distances, 

Vor. I. 66 ° ab, 


514 PROMISCUOUS EXERCISES. 


ab, bc, &c. on a thread, or cord, or chain, &c. having in itself 
little orno weight. For the equality of the weights, and their 
directions and distances, will put the whole line, when they 
come to equilibrium, into the same festoon shape or figure. 
So that. whatever properties are inferred in the coroilaries 
to the foregoing prob. will equally apply to the festoon or 
lower figure hanging in equilbrio. 

This is a most useful principle in all cases of equilibriums, 
especially to the mere practical’ mechanist, and enables him 
in any experimental way to resolve problems, which the best 
mathematicians have found it no easy matter to effect by mere 
computation. For thus, in a simple and easy way he obtains 
the shape of an equilibrated arch or bridge ; and thus also he 
readily obtains the positions of the rafters in the frame of an 
equilibrated curb or mansard roof ; a single instance of which 
may serve to show the extent and uses to which it may be ap- 
plied. Thus, if it should be required to make a curb frame 
roof having a given width ag, and 
consisting of four rafters aB, Bc, 
ep, DE, which shall either be 
equal or any given proportion to 
each other. Jhere can be no 
doubt but that the best form of 
the roof will be that which puts 
all its parts in equilibrio, so that there may be no unbalanced 
parts which may require the aid of ties or stays to keep the 
frame in its position. Here the mechanic has nothing to do 
but to take four like but small pieces, that are either equal 
or in the same given proportions as those proposed, and con- 
nect. them closely together at the joints a, B, c, p, £, by pins 
or strings, so as to be freely moveable about them ; then sus- 
pend this from two pins a, e, fixed 





ina horizontal line, and the chain ; L 

of the pieces will arrange itself Cries Hr § 
in such a festoon or form, abcde, / A 
that all its parts will come to rest if re 
in equilibrio.. Then, by invert- @¢-————-— 


ing the figure, it will exhibit the 

form and frame of a curb roof 

a€yde, which will also be in equi- b 

librio, the thrusts of the pieces — a 

now balancing each other, in the 

same manner as was done by the mutual pulls or tensions of 

the hanging festoona bec de. By varying the distance ae, of 

the points of suspension, moving them nearer to, or farther 

off, the chain will take different forms; then the frame apcpE_ 
may 


PROMISCUOUS EXERCISES. 515 


may be made similar to that form which has the most pleasing 
or convenient shape, found above as a model. 

Indeed this principle is exceeding fruitful in its practical 
consequences. It is easy to perceive that it contains the 
whole theory of the construction of arches: for each -stone 
of an arch may be considered as one of the rafters or beams 
in the foregoing frames, since the whole is sustained by the 
mere principle of equilibration, and the method, in its appli- 
cation, will afford some elegant and simple solutions of the 
most difficult cases of this important problem. 


PROBLEM 32. 


Of all Hollow Cylinders, whose Lengths and the Ditonigtér of 

" the Inner and Outer Circles continue the same, it 1s required 
to show what will be the Position of the Inner Circle when the 
Cylinder is the Strongest Laterally. 


Since the magnitudes of the two circles are constant, the 
area of the solid space included between their two circum- 
ferences, will be the same, whatever be the position of the 
inner circle, that is, there is the same number of fibres to be 
broken, and in this respect the strength will be always the 
same. The strength then can only vary according to the 
situation of the centre of gravity of the solid part, and this 
again will depend on the place where the cylinder must first 
break, or on the manner in which it is fixed. 

Now, by cor. 8 art. 251 Sta- 
tics, the cylinder is strongest {i 
when the hollow, or inner cir- { J 
cle, is nearest to that side 
_where the fracture is to end, 
that is, at the bottom when it 
breaks first at the upper side, 
or when the cylinder is fixed 
only at one end as in the first 
figure. But the reverse willbe 
the case when the cylinder is 
fixed at both ends ; and con- 
sequently when it opens first below, or ends above, as in the 
2d figure annexed. 





PROBLLM 33. 


‘To determine the Dimensions of the Strongest Rectangular Beam, 
that can be cut out of a Given Cylinder. 
Let 


B16 PROMISCUOUS EXERCISES. 


Let as, the breadth of the required 
beam, be denoted by 6, ap the depth by 
‘ d, and the diameter ac of the cylinder 
by p. Now when as is horizontal, the 
lateral strength is denoted by bd? (ty art. 
248 Statics), which is to be a maximum, 
But ap? = ac? — aB?, or d? =p? — 62; 
theref. 6d? = (p? 2) = == p2b—b3 is a 
maximum : in fluxions p?—362b=0=p? —36?, or p?=362; 
also d?==p? —b? =362 —6?==2b3. Conseq. b? : d? : np? :: 
1: 2: 3, that is, the squares of the breadth, and of the 
depth, and of the cylinder’s diameter, are to one another re- 
spectively as the three numbers 1, 2, 3. 

Corol. 1. Hence results this easy prac- 
tical construction : divide the diameter ac 
into three equal parts, at the points £, F ; 
erect the perpendiculars EB, Fp ; and join 
the points B, p to the extremities of the 
diameter : so shall ascp be the rectangu- 
lar end of the beam as required. For, 
because AE, AB, ac are in continued pro- 
portion (theor. 87 Geom.), theref. az: ac :: AB? : ac? : and 
in like manner aF : ac :: ap? : ac? ; hence aE: AF: 4c:: 
ABA ADEs packet Soke ee. 

-Corol. 2.. The ratios of the three 6, d, vp, being as the 
three .f 1, ¥ 2,4/3, or as 1, 1-414 1-732, are nearly as the 
three 5, 7, 8:6, or more nearly as 12, 17, 20°8. 

Corol.. 3. A square beam cut out of the same cylinder, 
would have its side = p4/ } 4==1p,/2. And its solidity would 
be to that of the strongest beam, as 1p? tojp? vy 2, or as 
3 to 2,/2, oras 3 to 2-898 ; while its strength would be to that 
of the strongest beam, as (p,/1)3 top ,/1X2p?, orasi 4/2 
to 24/3, or as 9 2 to 8,/3, or nearly as 101 to 110. 

Corol. 4. Either of these beams will exert the greatest 
lateral strength, when the diagonal of its end is placed verti- 
cally, by art. 252 Statics. 

Corol. 5. The strength of the whole cylinder will be te 
that of the square beam, when placed with its diagonal verti- 
cally, as the area of the circle to that of its inscribed square. 
For, the centre of the circle will be the centre of gravity of 
both beams, and is at the distance of the radius from the 
Yowest point in each of them; conseq. their cirepaiee will 
be as their arrears, by art. 243 Statics. 








PROBLEM 


PROMISCUOUS EXERCISES. 517 
PROBLEM 34. 


To determine the Difference in the Strength of a Triangular 
Beam, according as at lies with a Edge or with the Flat Side — 
Upwards. 


In the same beam, the‘area is the same, and therefore the 
strength can only vary with the distance of the centre of 
gravity from the highest or lowest point ; butina triangle, the 
distance of the centre of gravity from an angle, is double of 
its distance from the opposite side : therefore the strength of 
the beam will be as 2 to 1 with the different sides upwards, 
under different circumstances, viz. when the centre of gra- 
vity is farthest from the place where fracture ends, by art. 243 
Statics, that is, with the angle upwards when the beam is 
supported at both ends ; : but with the side upwards, when it 
is supported only at one end, (art. 252 Statics), because in the 
former case the beam breaks first below, but the reverse in 
the latter case. 


PROBLEM 35. 


Given the Length and Weight of a Cylinder or Prism, placed 
Horizontally with one end firmly fixed, and will just support 
a given weight at the other end without breaking ; it is requir- 
ed to find the Length of a Similar Prism or Cylinder, which, 
when supported in like manner at one end, shall just haor 
without breaking another given weight at the unsupported end, 


Let / denote the length of the given cylinder or prism, d 
the diameter or depth of its end, w its weight, and wu the 
weight hanging at the unsupported end; also let the like 
capitals L, D, w, v denote the corresponding particulars of 
the other. prism or cylinder. Then, the weights of similar 
solids of the same matter being as the cubes of their lengths, 


ests 13238 to, the weight of the prism whose length, 


is L. Now 1wi will be the stress on the first beam by its own 
weight w acting at its centre of gravity, or at half its length ; 
and lu the stress of the added weight w at. its extremity, their 
sum (1w-+w)/ will therefore be the whole stress on the given 
beam: in like quennien, the whole stress on the gine beam, 


3 one will be (Lw-tv)z or (Seu). 


But the lateral strength of the first beam is to that of the 
second, as d3 to n? (art. 246 Statics), or as 1? to 13 ; and the 
‘strengths and stresses of the two beams must be in the same 
ratio, “to answer the conditions of the problem: therefore as 


(Aw+tuyl 


whose weight is w or ~ 


518 PROMISCUOUS EXERCISES. 


—Go--u)l : ae tv) :: 19: 135 this analogy, turned into 


ea 


2 





an equation, gives 13 — ln aed ate 0, a cubic equa- 


tion from which the nuiieral value of L may be easily deter- 
mined, when those of the other letters are known. 
Corol. 1. When ’v vanishes, the equation gives L* = 


oa vay) 


Qu | 
lu?, orL = oss a whence w: w+2u :: 1: 1, for the 





ieneth of the beam, which will but just support its own 
weight. | 

Corol. 2. If a beam just only support its own weight, 
when fixed at. one end; thea a beam of double its length 
fixed at both ends, will also just sustain itself: or if the one 
just break, the other will do the same. 


PROBLEM. 36. 


Given the Length and Weight of a Cylinder or Prism, fixed 
Horizontally as in the foregoing problem, and a weight which, 
when hung at a given point, Breaks the Prism; tt is required 
to determine how much longer the Prism, of equal Diameter 
or of equal Breadth and Depth, may be extended before it 
Break, either by its own pees or by the addition of any other 
adventitious weight. 


Lal l denote the length of the given prism, w its weight, 
and wv a weight attached to it at the distance d from the fixed 
end; also let zt denote the required length of the other prism, 
and vu the weight attached to it at the distance p. Now the 
strain occasioned by the weight of the first beam is 1wl, and 
that by the weight u at the distance d, is du, their sum twl 
-+- du being the whole strain. In like manner }wr -++ pv is 


L 
the strain on the second beam; butl: 13 :: w: a= w the 


weight of this beam, theref. --+ pu = its strain. But the 

strength of the beam, which is just sufficient to resist these 

strains, is the same in both cases ; therefore + DU = 

4wl -- du, and hence, by r reduction, the required length 
wlt+2du— PRUs 


WL ee 
DW 
Corol. 1. When. the tallethened beam just brealcs by: its 
own 





hei bec EXERCISES, 519 


ewn weight, then vu = : 0 or vanishes, and the required length 
wl-2du 
becomes L = 4/ (UX ees: 
Corol. 2. Also when v vanishes, if d becomes = l, then 


oor meee 





is the required length. 


PROBLEM 37, 


Let as be a beam moveable about the end a, so as to make any 
angle Bac with the plane of the horizon ac : it is required to 
determine the position of a prop or supporter pE of a given 
length, which shall sustain wt with the greatest ease in any 
given position; also to ascertain the angle Bac when the least 
force which can sustain as, is greater than the least force in 
any other position. 


Let c be the centre of gravity of the 
beam ; and draw em perp. to aB, Gn to 
ac, nm tocm, and aru to pe. Putr= 
AG, p=DE, w = the weight of the beam 
aB, and an= ax. Then by the nature of 
the parallelogram of forces, Gn : «am, or_ 
by sim. triangles, ac==r: an =a2:: Ww: 


=, the force which acting at c in the di- 





rection me, is sufficient to sustain the beam's and by the nature 
of the lever, az : ac=r:: = the requisite force atc : =, 
the force capable of supporting it at & in a direction perp. to 
as or parallel to mo; and again as ar: aE:: = tes! the 


force or pressure actually sustained by the given prop DE in a 
direction perp. to ar. And this latter force will manifestly 
be the least possible when the perp. ar upon pz is the great- 
est possible, whatever the angle pac may be, which is when 
the triangle ape is isosceles, or has the side ap=akE, by an ob- 
vious corol. from the latter part of prob. 6, Division of Sur- 
faces, vol. 1. 

Secondly, for a solution to the latter part of the problem, 


WE . 
we have to find when — is a maximum: the angles p and & 
3 AF 


_ being always equal to each other, while they vary in magni- 
tude by the change in the position of az. Let ar produced 
meet en in H: then, in the similar triangles apr, arm, it will 

be 


520 PROMISCUOUS EXERCISES. 


be ar: an=x :: pF=ip: Hn, hence ry tos ip’ and conseq. - 
Hn 2 
X w=—Xw. But, by theor. 83, Geom. and comp. AG-F An 


Eee au 


=r+r: an=a2:: Gn=/ (r? ~z2) > Ha = 








r+x 

=1,/—— : consequently the force — xX w, acting on the 
re ~ wae BP r—a 

prop, is also truly expressed by —4/ . Then the flux- 
ip r+ex 


r the cos. angle 





bal 
Q 


ion of this made to vanish gives x= 
Bac==51° 50’, the inclination required. 


PROBLEM 38. 

Suppose the Beam az, instead of being moveable about the centre 
A, as tn the last problem, to be supported in @ given position 
by means of the given prop pE: it is required to determine 
the position of that prop, so that the prismatic beam ac on 
which it stands, may be the least liable to breaking, this latter, 
beam being only supported at its two ends a and c. 


Put the base ac = 6, the prop pr = p, 
Ae =r, the weight of an = w, s and ¢ the 
sine and cosine of Z a,x = sin ZE,y = 
sin. ZD, and z = ae. Then, by trigon. 

ps 


wl “338 
ZIyri pis, or 2 ==~, and an =; also 
ry x s 





cw = the force of the beam atc in direc- A Dw “© 
tionam. Let r denote the force sustain- j 

ing the beam at ein the direction Ep: then, because action 
and re-action are equal and opposite, the same force will be 


exerted at p in the direction pe: therefore ac .cw = F2x, 


and F = —. Again, the vertical stress at p, will be as F x 
& 


sine DXAD . DC=Fy. AD. Do= eal Pro) == (substi- 
2x ms 


: u ae bs ~ 
tuting — for its equal zy se shisd =rew XP 

p 2 OF s 8 Bos 

TCW bs og bs. 
—t x be — x) ==aminimum by the problem. Conseq. — 
—2X is a minimum, or 2 a maximum, thatis, s=1, and the an- 

gle eis a right angle. Hence the point © is easily found by 

this proportion, sin. A: COs. A: : ED: EA. ts 

Br Ra idck hee ) >) PROBLEM 


PROMISCUOUS EXERCISES. 521. 
PROBLEM 39. | 
To explain the Disposition of the Parts of Machines. 


When several pieces of timber, iron, or any other materi- 
als, are employed in a machine or structure of any kind, all 
the parts both ef the same piece, and of the different pieces in 
the fabric, ought to be so adjusted with respect to magnitude, 
that the strength in every part may be, as near as possible, in 
a constant proportion to the stress or strain to which they 
will be subjected. Thus, in the construction of any engine, 
the weight and pressure on every part should be investigated 
and the strength apportioned accordingly. All levers, for 
instance, should be made strongest where they are most 
strained : viz. levers of the first kind, at the fulcrum ; levers 
of the second kind, where the weight acts ; and those of the 
third kind, where the power is applied. The axles of wheels 
and pulleys, the teeth of wheels, also ropes, &c. must be. 
made stronger or weaker, as they are to be more or less act- 
edon. The strength allotted should be more than fully com- 
petent to the stress to which the parts can ever be liable ; but 
without allowing the surplus to be extravagant for an over 
excess of strength in any part, instead of being serviceable, 
would be very injurious, by increasing the resistance the ma- 
chine has to overcome, and thus encumbering, impeding, and 
even preventing the requisite motion: while, on the other 
hand, a defect of strength in any part will cause a failure 
there, and either render the whole useless, or demand very 
frequent repairs. | 


PROBLEM 40. 
Te ascertain the Strength of Various Substances. 


The proportions that. we have given on the strength and 
stress of materials, however true, according to the principles 
assumed, are of little or no use in practice, till the compara- 
tive strength of different substances is ascertained: and even 
then they will apply more or less accurately to different sub- 
stances. Hitherto they have been applied almost exclusive- 
ly to the resisting force of beams of timber ; though proba- 
bly no materials whatever accord less with the theory than 
timber of all kinds. In the theory, the resisting body is sup- 
posed to be perfectly homogeneous, or composed of parallel 
fibres, equally distributed round an axis, and presenting uni- 
form resistance to rupture. But this is not the case in a beam 
of timber : for, by tracing the process of vegetation. it is readily 
Von, IF. : 67 seen 


522 PROMISCUOUS EXERCISES. 


seen that the ligneous coats of a tree, formed by. its annual 
- growth are almost concentric ; being like so many hollow cy- 
linders thrust into each. other, and united by a kind of me- 
dullary substance, which offers but little resistance : these 
hollow cylinders therefore farnish the chief strength and re- 
_ sistance to the force which tends to break them. 

Now, when the trunk of a tree is squared, in order that it 
may be converted into a beam, it is plain that all the ligneous 
cvlinders greater than the circle inscribed in the square or 
rectangle, which is the transverse section of the beam, are 
-cut off at the sides ; and therefore almost the whole strength 
or resistance arises from the cylindric trunk inscribed in the 
solid part of the beam; the portions of the cylindric ceats, 
situated towards the angles, adding but little comparatively to 
the strength and resistance of the beam. Hence it follows 
that we cannot, by legitimate comparison, accurately deduce 
the strength of a joist, cut from a small tree, by experiments 
on another which has been sawn from a much larger tree or 
block. As to the concentric cylinders abovementioned, they 
are evidently not all of equal strength : those nearest the 
centre, being the oldest, are also the hardest and strongest ; 
which again is contrary to the theory, in which they are sup- 
posed uniform throughout. But yet, after all however, it is 
still found that, in some of the most important problems, the 
results of the theory and well-conducted experiments coin- 
cide, even with regard to timber: thus, for example, the ex- 
periments on rectangular beams afford results deviating but in 
a very slight degree from the theorem, that the strength is 
proportional to the product of the breadth and the square of 
the depth. 

Experiments on the Sikonsth of different kinds of wood, 
are by no means so numerous'as might be wished: the most 
useful seem to be those made by Muschenbroek, Buffon, 
Emerson, Parent, Banks, and Girard. But it will be at all 
times highly advantageous to make new experiments on the 
same subject ; a labour especially reserved for engineers who 
possess skill and zeal for the advancement of their profession. 
It has been found by experiments, that the same kind of wood, 
and of the same shape and dimensions, will bear or break 
with very different weights ; that one piece is much stronger 
‘than another, not only cut out of the same tree, but out of 
the same rod; and that even, if a piece of any length, planed 
equally thick throughout, be separated into three or four 
pieces of an equal length, it will often be found that these 
pieces require different weights to break them. Emerson 
ratesablas that wood from the boughs and branches of trees is 


far 


PROMISCUOUS EXERCISES. 593 


far weaker than that of the trunk or body-; the wood of the 

large limbs stronger than that of the smaller ones ; and the 

wood in the heart of a sound tree strongest of all; though 

some authors differ on this point. It is also observed that a 

piece of timber which has borne a great weight for a short 

time, has broke with a far less weight, when left upon it for 

a much longer time. Wood is also weaker when green, and 

strongest when thoroughly dried, in the course of two or 

three years, at least. Woodis often very much weakened by 

knots in it ; also when cross-grained, as often happens in saw- 

ing, it will be weakened in a greater or less degree, according 

as the cut runs more or less across the grain. From all 
which it follows, that a considerable allowance ought to be. 
made for the various strength of wood, when applied to any 

use where strength and durability are required. 


Iron is much more uniform in its strength than wood. Yet 
experiments show that there is some difference arising from 
different kinds of ore: a difference is also found not only in 
iron from different furnaces, but from the same furnace, and 
even from the same melting ; which may arise in a great mea- 
sure from the different degrees of heat it has when pons 
into the mould. 


Every beam or bar, whether of wood, iron, or stone, is 
more easily broken by any transverse strain, while it is also 
suffering any very great compression endways ; so much so 
indeed that we have sometimes seen a rod, or a long slender 
beam when used as a prop or shoar, urged home to such a 
degree that it has burst asunder with a violent spring. Se- 
veral experiments have been made on this kind of strain: a 
piece of white marble, } of an inch square. and 3 iaches long, 
bore 38lbs; but when compressed endways with 300lbs, it 
broke with 141 Ibs.. The effect is much more observable in 
timber, and more elastic bodies ; but is censiderable in all. 
This is a point therefore that must be attended to in all ex- 
periments ; as well as the following, viz. that a beam support- 
ed at both ends, will carry almost twice as much when the 
the ends beyond the-props are kept from rising, as when the 
beams rest loosely on the props. ! 


The following list of the absolute strength of several ma- 
terials, is extracted from the collection made by professor 
Robison, from the experiments of Muschenbroek and other 
experimentalists. The specimens are supposed to be prisms 
or cylinders of one square inch transverse area, which are 
stretched or drawn lengthways by suspended weights. gradu- 
ally increased till the bars parted or were torn asunder by the 

number 


Si PROMISCUOUS EXERCISES. 


number of avoirdupois pounds, ona medium of vii trials, 
set Peuposite each name. 
Ist Merats. airs 
Ibs. Ibs. 


Gold, cast ~.. °. 22,000" Tin, cast. 942. . 5,000 
Silver, cast .. . . 42,000 Lead, ort | menor tet 860 
Copper, cast . . 34,000 Regulus of Antimony 1,000 
{von. cast see 6s GO O00 - IMG ete ce 2,600 
fron; bars.) <3. 2. 70,080-* Bismuth... 6 2,900 


Steel, bars: 6s 135,000 

It is very remarkable that almost all the metallic mixtures 
are more tenacious than the metals themselves. The change of 
tenacity depends much on the proportion of the ingredients ; 
and yet the proportion which produces the most tenacious 
mixture, is different in the different'metals. The proportion 
of ingredients here selected, is that which produees the great- 
est strength. 


Ibs. a Ibs. 
2 parts gold with 1 Brie: of eopeeee and tin 51,000 
silver . 28,000 3tin, 1 lead . - 10,200 


5 pts gold, t copper 50,000 8tin, 1 zinc. . Re 10,000 
Ssilver, 1 copper . 48,500 4 tin, I regul. antim. 12,000 
4silver, 1tin . . 41,000 Slead, 1.zme . . 4,500 
6 copper, 1 tin . . + 60,000 4 tin, I lead, 1 zinc 13,000 
These numbers are of considerable use in the arts. The 
mixtures of copper and tin are particularly interesting in the 
fabric of great guns. By mixing copper, whose greatest 
strength does not exceed 37,000 with tin which does not ex- 
ceed 6000, is produced a metal whose tenacity is almost double, 
at the same time that it is harder and more easily wrought : 
it is however more fusible. We see also that a very small 
addition of zinc almost doubles the tenacity of tin, and in- 
creases the tenacity of lead 5 times ;:and a small addition of 
lead doubles the tenacity of tin.. These are economical mix- 
tures ; and afford valuable information to plumbers for aug- 
menting the strength of water-pipes. Also, by having re- 
course to these tables, the engineer can proportion the thick- 
mess of his pipes, of whatever metal, to the pressures they 
are to suffer. 


2d. Wusds: Sic. ; 


i BA ee Ibs... 4 Ibs. 
Locust tree yg CO AO0 MIE so ASOD 
Jujed wee yeh 6S 9855000" Mulbetry 0% 2 82 )500" 
Beech, Oak. ~ 17,800) Willow. 3 6)... 123500 
Orange .>% BS1600 «Wehbe npc ae 


Alder e * ° * > 13,900 Plum © ° ° ° e 11,800 


PROMISCUOUS EXERCISES. 525 
Elder °:csee seus 40,000 Cypress). 4-4. 6,000 
Pomegranate . . . 9,750 Poplar .....)>. .° 5,500 
Lemon Vipin: kee eeQ: Aoedar nine ac 4880 
Tamarind... 8,750 Ivory) .-°.0 6s 516,270 
Bara est be OOo S BONES Pi eae: be 3250 
Walnut Ue BTGO, Barneys Bae og 8,750 
Pitch. pine . 55. 3s ee Whalebone . . . 7,500 
‘Quince... 6,7 Tooth of sea-calf. . 4,075 
It is to be observed that ‘Ress numbers express something 
more than the utmost cohesion ; the weights being such as 
will very soon perhaps in a minute or two, tear the rods 
asunder. It may be said in general, that 2 of these weights 
will sensibly impair the strength after acting a considerable 
while, and that one-half is the utmost that can remain per- 
manently suspended at the rods with safety ; and itis this last 
allotment that the engineer should reckon upon in his con- 
structions. There is however considerable difference in this 
respect : woods of a very straight fibre, such as fir, will be 
less impaired by any load which is not sufficient to break them 
immediately. According to Mr. Emerson, the load which 
may be safely suspended to an inch square of various materi- 
als, is as follows : 
Ibs. Ibs: 
Wonk sc. a Te A00: Red fir, holly, HER 
Brags) i. -4. 0. 6, 08,000 DIANE ee eso 5,000 
Hempen rope - » 19,600 Cherry, hazle . . 4,760 
Ivory. <=. 15,700 Alder, asp, birch, | 
Oak, box, yew, plum 75050... willow. 203s gc 4,290 
Bim, ash, beech. . 6,070 Freestone. . . . 914 
Walnut, plumb aap fommegk ees Uy OY ts Nan eee Meee ee Ongeme os! 7 
 cwts. 
He gives also the practical rule, that Iron... . 135d? 
a cyhader whose diameter is d inches, Good rope. 22d? 
loaded to 1 of its absolute strength, will Oak .... 14d?’ 
carry permanently as here annexed. Pie rie os BOS 
Experiments on the transverse strength of bodies are easily 
made, and accordingly are very numerous, especially those 
made on timber, being the most common and the most inte- 
resting. The completest series we have seen is that given 
by Belidor, in his Science des Ingenieurs, and is exhibited in 
the following table. The first column simply indicates the ' 
number of the experiments ; the column 0 shows the breadth 
of the pieces, in inches ; the column d contains their depths ; 
‘the column / shows the lengths; and column /bs shows the 
weights in pounds which broke them, when suspended by 
their middle points, being the medium of 3 trials of each 
piece ; the accompanying words, fixed and loose denoting 


526 ' PROMISCUOUS EXERCISES. 


whether the ends were firmly - fixed down, or simply lay ee 
on the supports. 


No. |b d L- lbs. 

1 1. | Bee are It 406. loose. 
2 1 1 18 | 608 fixed. 
3 2, 1 18 805 loose. 
4 1 2g 18 1580 loose. 
oe I 1 |) 36 187 loose. 
6 1 1 36 283 . fixed. 
7 2 Q 36 1585. loose. 
8 12] 21°}. 36 1660 loose. 


By comparing experiments 1 and 3, the strength appears 
proportional to the breadth. 

Experiments 3 and 4 show the strength to be as the breadth 
multiplied by the square of the depth. | 

Experiments 1 and 5 show the strength nearly in the in- 
verse ratio of the lengths, but with a sensible deficiency in 
the Jonger pieces. 

Experiments 5 and 7 show the strength to be proportional 
to the breadth and the square of the depth. 

Experiments 1 and 7 show the same thing, compounded 
with the inverse ratio of the length; the deficiency of which 
is not so remarkable here. 

Experiments | and 2, and experiments 5 and 6, show the 
increasé of strength, by fastening down the ends, to be in the 
proportion of 2 to 3; which the theory states as 2 to 4, the 
difference being probably owing to the manner of fixing. 

Mr. Buffon made numerous experiments, both on small 
bars, and on large ones, which are the best. The following 
is a specimen of one set, made on bars of sound oak, clear of 
knots. : 
Broke 


Length|Weight| with | Bent | Time 
| feet, Ibs. Ibs. | inch. | min. 

















60 | 5350-| 3:5 |. 29’ 
; 56 |} 5275 | 4:5 |. 22 
68 | 4600 | 3-75 | 15 
63 | 4500 | 4-7 | 13 
9 77 | 4100 | 4:85 | 14 . 
V1.1. 3950.1 5-5 as fee thy 
(84 | 3625 | 5:8 ieee 
; 82.| 3600 |} 65 | 15 
100 |. 3050 | 7 | 
98 | 2925 | 8 





ru Rl le ae Sieh 


PROMISCUOUS EXERCISES. 524 


Column 1 shows the length of the bar, in feet, clear be- 
tween the supports,—Column 2 is the weight of the bar in 
Ibs, the 2d day after it was felled.—Column 3 shows the num- 
ber of pounds necessary for breaking the tree in a few mi- 
nutes.—Col. 4 is the number of inches it bent down before 
breaking.—Col. 5 is the time at which it broke.—The parts 
next to the root were always the heaviest and strongest. 

The following experiments on other sizes were made in the 
same way ; two at least of each length being taken, and the 
table contains the mean results. ‘The beams were all squared, 
and their sides in inches are placed at the top of the columns, 
their vanes e in feet being in the first column. The numbers 






















































in the other columns, are the ae weight which broke the 
pieces. 2 
4. car es a eae bees ny | 
SS a 
of 7 46312)11525 118950 =e 47649 ||11525 | 
| 8 sae 115525 |26050 |39750 Wrage | 
9 |4025; 8308 113150 122350 |32800 | 8964. | 
10 |3612| 7125 Hse 19475 ey 3068 | 
12 |2987| 6075 | 9100 116175 |23450 [e733 
14 | 5300 | 7475 jiseen 19775 || 5763 
16 14350 | 6362 |11000 |16375 | soxe 
18 3700. | 5562 | 9245 113200 || 4482 
20 | 3225 | 4950 * 8375 |11487 | 4034 | 
| 22 | 2975 3667 | 
24 2162 | 3362 
| 28 | 1775 | 2881 | 


. Mr. Buffon had found, by many trials, that oak timber lust 
much of its strength in the course of seasoning or drying ; and 
therefore, to secure uniformity, his trees were all felled in — 

the same season of the year, were squared the day after, and 
the experiments tried the third day, ‘Trying them in this 
green state gave him an opportunity of observing a very cu- 
rious phenomenon. When the weights were laid quickly on, 
_nearly sufficient to break the beam, a very sensible smoke 
was observed to issue from the two ends with a sharp hissing 
sound; which continued all the time the tree was bending 
and cracking, _ This shows the great effects of the compres- 
sion, and that the beam is strained through its whole length, - 

which is shown also. by its Randne through the whole 
length. | 
Mr. Buffon considers the experiments with the 5-inch bars 
as the standard of comparison, having both extended these to 
greater lengths, and also tried more pieces of each Bs 
NOW, 


528 PROMISCUOUS EXERCISES. 


Now, the theory determines the relative strength of bars,,of 
the same section, to be inversely as their lengths: but most 
of the trials show a great deviation from this rule, probably 
owing, in part at least, to the weights of the pieces them. 
selyes. Thus, the 5-inch bar of 28 feet long should have 
half the strength of that of 14 feet or 2650, whereas it is only 
1775; the bar of 14 feet should have half the strength of that 
of 7 feet, or 5762, but is only 5300; and so of others. The 
column a is added, to show the strength that each of the 5-inch 
bars ought to have by the theory. _- 

Mr. Banks, an imgenious lecturer on natural philosophy, 
has made many experiments on the strength of oak, deal, and 
iron. He found that the worst or weakest piece of dry heart 
of oak, 1 inch square, and 1 foot long, broke with 602Ibs, and 
the strongest piece with 974lbs : the worst piece of deal broke 
with 464lbs, and the best with 690lbs. A like bar of the 
worst kind of cast iron 2190lbs.. Bars of iron set up in posi- 
tions oblique to the horizon, showed strengths nearly propor- 
tional to the sines of elevation of the pieces. Equal bars 
placed horizontally, on supports. 3 feet distant, bore 62, cwt ; 
the same at 2} feet distance broke only with 9 cwt.—An arch- 
ed rib of 293 feet span, and 11 inches high in the centre, sup- 
ported 991 cwt; it sunk in the middle 37 inches, and rose. 
again 2 on removing the load. The same rib tried without 
abutments, broke with 55 cwt.—Another rib, a segment ofa 
circle, 291 feet evel and 3 feet high in the middle bore 1002 
cwt, and sunk 1,3, in the middle. The same rib without abut- 
meuts, broke with 642 cwt. 

Mr. Banks made also experiments at another foundery, on 
like bars of 1 inch square, each yard in 1 length weighing 9lbs, 
the props at 3 feet asunder. 

The Ist bar broke with . ..... . = + 963 Ibs. 

The 2d ditto. . . ore kee Bone) et Oe 

The od Otte oo sa ac hus 4s ge ee 

Bar made from the cupola, broke with . . 864 

Bar equally thick in the middle, but the ends 

shaped into a parabola, and weighed 6,3, los, 

broke with . . 874 - 
From these and many other experiments, “Mr. Banks con- 
cludes, that cast iron is from 31 to 42 times stronger than oak 
of the same dimensions, and from 5 to 62 times stronger than 
deal. 


Some Examples for Panties, 


The theory, as has been before mentioned, is, That the 


strength of a bar, or the weight it will bear, is directly as 
oe the 


_ PROMISCUOUS EXERCISES. 529 


the breadth and square of the depth divided by the lengths 
So that, if 6 denote the breadth of a bar, d the depth, J the 
length, and w the weight, it will bear; and the capitals x, p, 
L, w denote the like quantities in another bar; then, by the 
bd2 ~ BD? 

rule eS ere ON which gives this general equation 
bd? uw = sp2lw, from which any one of the letters is easil y 
found when the rest are given. 

Now, if we take, for a standard of comparison, this expe- 
riment of Mr. Banks, that a bar of oak an inch square and a 
foot in length, lying on a prop at each end, and its strength, 
or the utmost weight it can bear, on its middle, 660lbs ; here 
6=1,d=1,1=1,w=660; these substituted in the above 
equation, it becomes Lw = 660sp?, from which any one of 
the four quantities 1, w, B, p, may be found, when the other 
three are given, when the calculation respects oak timber. 
But for fir the like rule will be tw = 400sp?, and for iron 
Lw=2640pp?. 

Exam. 1. Required the utmost strength ‘of an oak beam, of 
6 inches square and 8 feet long, supported at each end, cr the 
weight to break it in the middle ? TW Lo) Seon Ee 

Here are given 8=6, p= 6, L = 8, to find w = Ge Tee 


aad ani =660 X3X9=178201bs. 


Exam. 2. Required the depth of an oak beam, of the same 
length and orate as above, but only 3 inches breadth ” 

Here, as3.: 6 :: 36: p?=72, theref. PM ote ee the 
depth. 

This last beam, though as strong as the former, is but little 
more than 2 of its size or quantity. And thus, by making 
" joists thinner, a great part of the expense is saved, as in the 
modern style of flooring, &c. — 

‘Exam. 3. to determine the utmost strength of a deal joist 
of 2 inches thick and 8 inches deep, the bearing or breadth of 
the room being 12 feet ?—Here s=2, p==8, L=12; then the 


A40XBxXD? 440X264 
rule tw = 4408p? gives w= —— ome! 


iho = 4693 lbs, 


Exam. 4. Required the dears of 2 a bar of iron 2 inches 
broad and 8 feet long, to sustain a load of 20,000Ibs ?—Here 
=2, t= 6, and w = 20,000, to find p from the equation LW 


==26408p?, viz. p? == va _, 8% 20000 ee ee 30:3, and 
2) 2640p. 2640%2- = 38 
p=,/30°3==51 inches the ae 
Von, Tf, . Aram, 





2a 
eee a nen 


L 12 





. 680 PROMISCUOUS EXERCISES? 


Exam. 5. To find the length of a bar of oak, an inch 
_ Square, so that when supported at both ends it may just break . 
by its own weight ?—Here according to the notation and cal- 
culation in prob. 36, /=1, w=2 of a lb, the weight of 1 foot 
ao 





in length, and u=660lbs. Thenit=1,/ 
57-45 feet nearly. 

Exam. 6. To find the length of an iron ia an inch square, 
that it may break by its own weight, when it is supported at 
both ends.—Here as before /=1, w= 3lbs nearly the weight 
of 1 foot in length, also wu = 2640. Therefore 1 =1 4/ 
w+tQu ; 


= 4f 3301= 





=41:97 feet nearly. 


Note. It might perhaps have been supposed that this last 
result should exceed the preceding one: but it must be con- 
sidered that while iron is only about 4 times ree than 
oak, it is at least, 8 times heavier. 

Exam. 7. When a weight w is suspended from © on the 
arm of acrane aBcpDE, it is required to find the pressure at 
the end p of the spur, and that at B paint the upright post ac. 


Here, by the nature of the lever — w= BD C 


the pressure at p in the vertical direction 
DF: but this pressure in pr is to that in ps 
: . CE. CE. DB 
as DF tO DB, VIZ. DF : DB : : —-w: —-——_ W 
| met cD DF.cD 
the pressure in pp ; ents again, DB : FB OF 


CE.DB_ CE : 
cb ::——---w : —- w=—-w the pressure against B in direc-' 
DP CD epRe i BC 


tion FB. : 
Thus, for example, if cn=16 feet, nore cp=8, sp =10, 


cE. wae 3 %3=10 tons for the 


fs 
pressure on the spur pB. Also —w aig 3 = 8 tons, the 





and w=3 tons ; then — 





force tending to break the bar ac at B. 
, PROBLEM 41. 
To determine the circumstances of Space, Penetration, Velocity, 


and Time, arising froma Ball moving with a Given Velocity 
and Striking a moveable’ Block fat Wood, or other Substance. 





PROMISCUOUS EXERCISES. 53h 


Let the ball move in the direction az passing through the 
centre of gravity of the block 8, impinging on the point c ; 
and when the block has moved through the space cp in con- 
sequence of the blow, let the ball have penetrated to the 
depth pe. 

Let 8 = the mass or matter in the block, — 

6 = the same in the ball, 

s = cp. the space moved ‘by the block, 

x = pe the penetration of the ball, and theref. 
s + x = ce the space described by the ball, 

a = the first velocity of the ball, * 

v = the velocity of the ball at r, 

u = velocity of the block at the same instant, 
¢ = the time of penetration, or of the motion, 
7 = the resisting force of the wood. 


Then shall ~ be the accelerating force of the block, 


and i the retarding force of the ball. 


_ Now because the momentum Bu, communicated to the 
block in the time 7, is that which is lost by the ball, namely — 
— by, therefore pz = — bv, and su = — bv. -But when 
v=a,u=0; therefore by correcting, Bu = b(a —v); or 
the momentum of the block is every where equal to the mo- 
mentum lost by the ball. And when the ball has penetrated 
to the utmost depth, or when u = v, this becomes Bu = 6 
(a -4u,) or ab =(8-+6)u; that is, the momentum before 
the stroke, is equal to the momentum after it. And the ve- 
locity communicated will be the same, whatever be the re- 

sisting force of the block, the weight being the same. 
Agrs 2 
, and — 

B 





Again, (by prob. 6, Forces), it is u? = 


03 = a X (s+ 2), or ee hieby by correction, a? — v2 = 
b(a? —v?)— —4grs p 

4a COE 
And when v =u, by ape u for v, and Bu? for 4grs, 


are 


4 
- -(s-+ a). Hence the pefelimion or’ = 


the greatest penetration bocumeac 2 ; and this again 


by writing ab for it its value (n-++-b)u, ies the greatest pene- 


Boa? ba? 
ion x aie — ——). Which is barel 
tration + = igr(e Eb) 7s x (1 aware hich is barely 


; ba? Beaten, 
equal to 7 when the block is fixed, or infinitely great ; and 
°o '- 
is 


§32 PROMISCUOUS EXERCISES, 


is always very nearly equal. to the same when B ‘is very 


ereat in respect of 6. 





HS a? b2 
- a? —u? oO) (eb)? _. Bt-+-2B8b., a7b 
Hence $+ t= Agr b= Agr ~ (B-+b)? Bc 
And theref.p -+6b:3 + 2b:: a rede, or ed : bis 
and s = se —=s pal 


Pea 4gr(p+b)2" 
Exam. When the ball is iron, and weighs 1 pound, 
penetrates elm about 13 inches when it moves with a veloc 
ty of 1500 feet per second, in which case, 


3 2 2 5 
Pg Oe ae ean erly 
b” 4gr  4X16),X33  193%13— 
coh 4808 


When B = 500Ib, and 6 = 1; thenu = pater ee Sar =x 


feet nearly pe second, the velocity of the block. 


Also 5 = 2% x= —_200*8 


4gr 4X%162 4 X32284 ~ 46121 
&, of an inch, which is the space moved by the block when 
the ball has completed its penetration. 








part of a foot, or 





Qs = 1 
Ss Ss, ee SS a: Lr 
‘And ¢ a etx G9 Part of a second, 0 
: Q 2, 2% Lee 
ROL 4614 12 6+13.231 1 ok aah 
ait: “VY600 <' 6. 281, 100. 6922 te 


second, the time of penetration. 


- PROBLEM 4%. 


‘lo find the Velocity and Time of a Heavy Body desteabiag 
down the “Airc of a Circle, or vibrating an - the Airc oy a Line 
‘fixed in the Centre. 

Let p be the beginning of the descent, | 
© the centre, and a the lowest point of the | 
circle ; draw pe and Pq perpendicular to 
-ac. Then the velocity in e being the same — 

as in a by falling through ne, it will be - 
| 0=24/(g XEQ)= By po ee 

TE=AQ. | 


But the Boe. bfibe time fis= 





tx 
Vf ov (2ra—2*) 27) 
slept 


4/ Af (2ra— a? Ka (a= xv (aa) 


B ond Ap = 





| , ieee radius ac. Theref.j t= =X 


PROMISCUOUS EXERCISES. | 533 


te a ee eer RGIS ES UTS: Pea 


16 Tanai) x (d=) eve 2) x4/(1 





=) 
where d= Or the diameter. 

._ =/fd at (gy. 5a 
ChAT ae ¢ yt ames as tate Ae 4, 6de 


&c.), by developing vl -)i in a series. 


But the fluent of 





mene arc to radius 34 and 


een)" 


Qe ; 
vers. x, or it is the arc whose rad. is 1 and vers. a which 


call a. . And let the fluents of the succeeding terms, without 
the coeflicients, ben, c,p, £, &c. Then will the fiuxion of any 
one as Q, at n distaves trae A, be Q = 2%, = xp, which 
suppose also = the flux. of bp — dz™—1// Supe — «?) = bp — 


sax — x? ’ 
d(n—1)ican"—2 / (ax — 22) — dcx"? x= — Hapeat Reahlahiea 
(n— fax"! —nx™ 


dieX — — = bp — d(n—3)ar + daze. 


























af (Ue ~~ 2? ) iS 
pence Py, Pst the coefficients of the like terms; 
— 1 on— | — Dyn—I ee 
a=}, hese a; and @ =O" - lar — 23 Qe BE cen oul 
2n ee | 
: 3 : : fa 
Which being substituted, the fluential terms become ae 
1 aa—2,/(oxr—a?). 1.3  3an—22,/(ac-2?) 
(rArgas 2 laa" Agee eae 
1.3.5 5ac—2x*,/(ar—2x?) | suas 
paris: .—— ~~‘ &c), O : 
a a Bite &c). Or the same fluents 


will be found by art. 80 Fluxions. 


3°1416,/d. 

But when, z = a, those terms become barel pees 

17a . 32a? 12), 32, 5243 ‘ : 
Ces 92d a2 did? 82, 42 , 6208 — &c.) ; which being 
subtracted, and « taken = ‘s there arises for the whole 
‘time of descending peel Be or the cornerted value of t = 

3: at 2.3242 12.32.52 
ge ear’ d2q2 | 92.42, “peat & ): 


When the arc is small, as in the vibration of the pendu- 
tum of a clock, all the terms of the series may be omitted 
after the second, and then ie time of a semi-vibration ¢ is 

k 5708. : 
nearly ="—"—=.7 5 X (1+), And theref. the times of 


vibration 


534 PROMISCUOUS EXERCISES. 


vibration of a pendulum, in different arcs, are as 8r -- a, or 8 
times the radius added to the versed sine of the arc. 

If p be the degrees of the pendulum’s vibration, on wo 
BY. of the feet point of the small arc, the radius being 7, 
the aiameter d, and 3:1416 =p: then is the length of that 


D dp 
arc A ge Oe ROY shad But the versed sine in terms of the 


180. 360° 
bd oe ee oe Theref 
arc is a = = — Seca Co, Fm ieee a5 yg oy herefore 
a eh A2 A4 vi EDs p*p4 i {pee 
do ade Se0r Seeger 
the first term, by rejecting all the rest of the terms on ac- 
RH! Bag D2: 
ount of tl ll = — ly = ——... Thi 
count of their smallness, or > = < nearly (aiei his 


value then peme substituted for = or = in the last near 
2r 


1°5708 


value of the time, - ie dcales t =— V5 x (i+ aap) 


nearly. And therefore the times of vibration in different 
small arcs, are as 52524-+-p?, or as 52524 added to the sqnare 
of the number of degrees in the arc. 

Hence it follows that the time lost in each second, by vi- 

at oe - : Peers DR 

brating in a circle, instead of the cycloid, is cotay and con- 
sequently the time lost in a whole day of 24 hours, or 24 % 
60 x 60 seconds, is $ p? nearly. In like manner, the seconds 
lost per day by vibrating in the arc of A degrees, Is 8 A?. 
Therefore, if the pendulum keep true time in one of these 
arcs, the seconds lost or gained per day, by vibrating in the 
other, will be § (oD? — A*). So, for example, if a pendulum 
measure true time in an arc of 3 degrees, it will lose 112 se- 
conds a day by vibrating 4 degrees ; and 262 seconds a day by 
vibrating 4 degrees; and so on. 

And in like manner, we might proceed for any other curve, 
as the ellipse, hyperbola, parabola &c. 

Scholium. By comparing this with the results of the prob- 
Jems 13 and 14, Prac. Ex. on Forces, it will appear that the 
times in the cycloid, and in the arc of a circle, and in any chord 
of the circle, are peers as the three quantities. 


1, 1+ igi abd poste. 1354 


er bees as the three ninenfiues 1,1 += —, 1°27324 ; the first 


and last being constant, but the middle one, or the time in the 
circle, 


PROMISCUOUS EXERCISES. 2 Bg 2 


circle; varying with the extent of the arc of vibration. Also 
the time in the cycloid is the least, but in the chord the 
greatest ; for the greatest value of the series, in this prob. 
when a = +7, on the arc ap is a quadrant, is 1°18014; and in 
that case the proportion of the three times is as the numbers 
1, 1:18014, 1-27324. Moreover the time in the circle ap- 
proaches to that in the cycloid, as the arc decreases, and they 
are very nearly equal when that arc is very small. 


PROBLEM. 43. 

To find the time and Velocity of a Chain, consisting of very smalt 
links, descending from a smooth horizontal plane ; the Chain 
being 100 inches long, and one inch of it hanging of the Plane 
at the commencement of Motion. 


Put a = 1 inch, the length at the beginning ; 
~ = 100 the whole length of the chain ; 
« = any variable length of the plane. 
wey a is the motive force to move the body 


and — oo =f the accelerative force. 





' : @9 
Hence vw >= *afs =2g x7 x. a ae ‘i 
Q ny 
The fluents give v2 = ee, But » = 0,-when x = a, 


sek £2 — O 
theref. by correction, v2 =2¢ X i 
the velocity for any length xz. And whet the chau a 
quits ei! pi x = 1, and then ng sereatent velocity is. 
ute oe Ey cs 386 X9999__ 
100.4.22:9% 
196-45902 Sache. or ie 871585 feet) per second. 
x 
Again ¢ OF = of 5 5X Beye Ra ai) : 


Pati A ved 
which is ¢ = 4/ 2g X log. BiG cue the time for any — 











the correct fluent of 


length « And when «=I 100, it is ¢= oh X log. 


v5 386 
100 9999 
ea ale = 2°69676 sétonds. the time sehen the last of 


the chain just wuts the plain. 


PROBLEM 44. | 

‘To find the Time and Velocity of a Chain, of very small Links, 
quitting a Pulley, by passing freely over it: the whole Length 
being 


> 


7 
536 PROMISCUOUS EXERCISES. 


being 200 Inches, and the one End hanging g 2 Inches below the 


other at the Beginning. ne 
Put a= 2, | = 200, and = sp any variable © A” 
difference of the two parts aB, ac. isan: | 


==. f, and wv or 2g fs : 4 ae te Le 





Hence the correct fluent i is v2 = =x j opand 


the general expression forthe _ 





¥ eloc. And when ae l, or when c arrives at Ay 16 S28 oa 


. eens ar as ; 
isos J Ex" =v VAIS age | 
1002 os fe 386 X 9999 
3 a Ra ee oa So fi 
/ (386 X — 5) = a *186-46902, uP 
inches, or 16371585 feet for the greatest veloci- : L 
ty when the chain just quits the pulley. 5 


Again, ¢ or — re ad es (at aay’ And the cor- 


he ET 
rect fluent is t= —— J X log. sh ae — a?) 


se sla for the Gee; And wrlien fi = 1, ‘it becomes ¢ = 
taldigae =a?) 200, Lal e) 





——*, the | general 


= Xlog = /- 
ee, 100-4 9999 PAS 
Vv 386 X Io og. - te = 2°69676 es the whole 


time when the chain sai quits the pulley. 

So that the velocity and time at quitting the pulley i in: this 
prob. and the plane in the last prob. are the same; the dis- 
tance descended 99 being the same in both. For though 
_ the weight / moved in this latter case, be double of what it 

was in the former, the moving force « is also double, because 
here the one end of the chain shortens as much as the other 

end lengthens, so that the space descended tx is doubled, 


and becomes x; and hence the accelerative force > of f is 


the same in both; and of course the velocity and time the 
same for the same distance descended. 
. PROBLEM 45. 

Lo find the Number of Vibrations inade by two Weights, connect- 
ed by u.very fine Thread, passing freely over a Tack or.a Pul- 
ley, while the less Weight is drawn up to it by the Descent of 


the heavier pence at the other End. 
Suppose 


PROMISCUOUS EXERCISES. 537 


Suppose the motion to commence at equal dis- 
tances below the pulley ats ; and that the weights 
are 1 and 2 pounds. 

Put a = as, half the length of the thread ; 

6 = 39! inc. or 322 feet, the second’s pend. 
x == BW = BW, any space passed over ; We 
z = the number of vibrations. : 






w—w B 
Then Ss a f =1 is the accelerating force.  -w 
a 


And hence v or,/4gfs = V/ Agfx, and ¢ r= Agfx 


he 
But by the nature of pendulums, a/, (ate) : 4/6 3: Avibrs3 
J the vibrations per second made by either weight, 


namely; the longer or shorter, according as the upper or 
under sign is used, if the threads were to continue of that 
ength for one second, Hence, then, as : 
Ie 6 Se cat Bee 
Risk a Wenvity ee: v def * ork) 
the fluxion of the number of vibrations. 
Now mune the upper sign + tahes place, the duent 1g 


Ay aah Pi a sae A at 


And oe “ =a, the same then becomes z = ,/ — X ie 

3b 1172 

1+ /2=/ — xlog.1 + 2= Ee 

‘688511, the whole number of wibratens ik ae the de- 
scending weight. 

But when the lower sign, or ~, bitsos place, the duent} is 

















b 
Jf aah X arc to rad. 1 and vers. = Which, when x = a, 


3X 39% 3°1416 W7F 
Nives Ip 4/8: 1416 X / 7553 > 3 Sie AL ee Cres 


1-227091, the whole number of vibrations made by the lesser 
or ascending weight. 

Schol. It is evident that the whole number of vibrations, 
in each case, is the same, whatever the length of the thread. 
is. And that the greater number is to the less, as 1:5708 to 
the hyp. log. of 1 +  / 2. 

Farther, the number of vibrations performed in the same 
time ¢, by an invariable pendulum, constantly of the same 


length a, is 4/7 —: = +781190. For the time of descending 
Yoru, UU, | 69 ; the 


538 PROMISCUOUS EXERCISES. 


the space a, or the fluent of ¢= —~—, when = 4, is to 
! | 4g fx ; 


J beh And, by the nature of pendulums, 4/ a: of b:: 1 
vibr. : 4/ . the number of vibrations performed in 1 second ; 


hence 1”: ¢:: vo: t hee: va the bel apoine number of vi- 
brations. | ) | 

So that the three numbers of vibrations, namely, of the 
ascending, constant, and descending pendulums, are propor- 
tional to the numbers 1:5708, 1, and hyp. log. 1 ++ 4f 2, or as 
15708, 1, and -88137; whatever be the length of the thread. 


REMARK. 


The solution here given by. Dr. Hutton to this 45th prob- 
lem, is erroneous ; one of his errors in the solution consists 
in his not attending to the difference of tension in the pendu- 
lum as it ascends, descends, or continues of an invariable 
length ; his method will give vibrations to the descending 
pendulum, even when the tension is infinitely small or nothing. 
A tfue investigation of the problem affords several curious 
results ; but sa some cases we are led to very tedious compu- 
tations. 


PROBLEM 46, 


To determine the Circumstances of the Ascent and Descent of 
two. unequal Weights, suspended at the two Ends of « Thread 
passing over a@ Pulley: the Weight of the Thread and of the 
Pulley being considered in the Solutron. 


Let J = the whole length of the thread ; ORs 
= the weight of the same ; | 
== aw the dif. of lengths at first ; 
: = w—w the dif. of the two weights ; LA 
e = aweight applied to the circumference, 
such as to be equal to its whole wt. and 
friction reduced to the circumference ; ew 
s== w-++w--a-+c the sum of the weights moved. 
Then the weight of 6 is a and ate 
first. But if « denote any variable space descended by w, or 
ascended by w, the difference of the Jengths of the thread 
will be altered 2x ; so that the difference will then be 6—2z, 
b—2Qx 7 
t 


is the moving force at 





and its weight a; conseq. the motive force there will be 


d— 


PROMISCUOUS EXERCISES. 539 


as befs, allrenich Se, ad ther ef. Brett ayy the 


accelerating force there. Hence then vv = 2gfx = 22x X 

dl -— en ai dl - ahb--ax 

, sl sl 

orv=2 bye 4 % / (ex-+-ax?) the general expression for the 
—a 


























ave fluents of which give v? =4g7x. 





velocity, putting e=- And when «= b, or w becomes 
as far below was it was above it at the beginning, it is barely 
v==2,/ bas for the velocity at that time. Also, when a, the 
weight of the thread, is nothing, the velocity is only 2 4/f 
= as it ought. 

' Again, for the time i or Se eV aa au Rie TERE ees Heat)? ; the flu- 
ents of which give ¢ = ay= og EVES the gene- 


ral expression for the ime of pea 7 space x. . 
And if the radicals be expanded in a series, and the 108: of 
it be taken, the same amg will eas 
32 
= VEX Ina aX 0-3 ée + aoe ©) | 


Which therefore becomes barely iam ag when a, the gg of 
the thread, is nothing ; as it ought. 


PROBLEM 47, 


Zo find the Velocity and Time of Vibration of a small Weight, 
Jixed to the middle of a Line, or fine Thread void of Gravity. 
‘and stretched by a given Tension ; ; the extent of the Vibration 
heing very small, 





abies ac half the length of the thread ; 
a = cp the extent of the vibration; 
&% = CE any variable distance from c; 
* «== wt of the small body fixed to the middle ; ; 
w=a 


540 PROMISCUOUS EXERCISES. 


w =a wt. which hung at each end of the thread, will 

be equal to the constant tension at each end, ae 

- in the direction of the thread. 

Now, by the nature of forces, ar : ce :: w the force in 
direction za : the force in direction ec. Or, because ac is 
nearly = ac, the vibration being very small, taking ac in- 
stead of az, It 18 AC: CE: ::W: 7 the force in Ee arising 
from the tension in =a. Which will be also the same for 


. 2w , : 
thatin es. Therefore the sum is aa the whole motive 


H 
force in Ec arising from the tensions on both sides. » Conse- 





2 3 
quently == = f the accelerative force there. Hence the 


—4Agwar 








equation of the fluxions ov OF Qefs Sig ae nd the flus. 

?=— ve . But when xz = a, this p~ ee, and should 
. ares 2 

be = 0; theref. the correct fluents are v? = 4gw x= ri o — 


a? owe 2 ; 
and hence v = ,f (4gw X Ae rm, the velocity of the little 


body w at any pointe. And when x = 0, it is v = 2a Ve 


for the greatest velocity at the point c. 
Now if we suppose w= 1 grain, w = 5lb troy, or 28800: 
prains, and 2/= «ss =3 feet; the velocity at c becomes 


1 X28 
SHIA ye So that 


if a = j, inc. the greatest veloc. is 5, ft. per sec. 
if a = 1 ‘inc. the greatest veloc. is ooh 7 ft, per sec. 
ifa@ = 6 inc. the greatest re is ane - ft, per sec. 


i a 
To fmd the time ¢, it is ¢ or = =a w # 


- *% f(a? “i ies 
Hence the correct fluent is ¢ = } L/— X arc to cosine — — and 
radius 1, for the time in DE. ren when « = 0, the hol 


psa ieee l 
time in po, or of half a vibration, :7854 ./ ac and conseq: 


the time of a whole vibration through pd is 1-5708 pees 


Wee 
Using the foregoing numbers, namely » = 1, w= 28800, 
and 


PROMISCUOUS EXERCISES. 541 


Rag La : ; 11112 
und 2] = 3 feet; this expression for the time gives ae = 


353%, the number of vibrations per second. But if w = 2, 
there would be 250 vibrations per second ; and if w = 100, 
there would be 35} vibrations per second. 


PROBLEM 48. 


Yo determine the same as in the last Problem, when the Distance 
cp bears some sensible Proportion to the Length an; the Ten- 
sion of the Thread however being still supposed a Constant 
Quantity. 





Using here the same notation as in the last problem, and 
taking the true variable length ar for ac, itis aE or EB: CE:: 
y 2w ; 

Ww :— = Sas the whole motive force from the two 

ual tensions w in az and gp ; and theref. a : =f 
eq ? $ ae afte +22 ) 

is the accelerative force at. Theref. the fluxional equation 


Swg 


— 07 : 
= ;-and the fluents v? roe x 


Ee a ees 
“TE +a) 

. 8wg 
— 4/ (12--+ 22). But when « = a, these are 0 = a ea ig 


saat Aiea. 
IS vy or 2efs = — 


o/ (l2 -+- a) ; therefore the correct fluents are v? =—3x 


[Y (2 + a?)— Y/Y (2 +22)|= re X (ap ide AE). And 
hence v = ,/ [8 x (ap —AzE)] the general expression for 
the velocity at c. And when & arrives atc, it gives the 
greatest velocity there =,/ pas X(ap—ac)]. Which when 
w=28800, w=1, 2/=3 feet, and co=6 inches or 1 a foot, is 
VW (8 X 28800 X 165, X Bat = 5483 feet per second. 


Which came out 555,7, in the last problem, by using always 
ac for age in the value of f. But when the extent of the vi- 
brations 


548 - PROMISCUOUS EXERCISES. 


brations is very small, as ;'5 of an inch, as it commons us 
at; greatest velocity ‘here will be of 8 X 28800 X 1675 


gzeo7= 9) nearly, which in the last problem was 9,'5 cee 


oe —L my) 
To find the time, it is 7 or ee x. tfc of Woe a? ))’ 


making c = ap=,/(l?'-+ a?). To ey the vaients the easier, 
multiply the numer. and feneh both by of [e+./(?-+2?),]} 


S0 shall ¢ = 4/ saat A arabe ~X Vf[et / (i? +27)]. 
Expand now the Sciuitity o/ b + J (3 t seit ina nae 
and put d=c-+1, so shall 7 =V/ STE (1 +75 


2d +l, Ade teditie iiat Peanone hen 
320718” 123a315 2048d417 








Bl se) 


2 ae 
Now the fluent of the first term ee —is = the arc to sine— 
2 am G2 a 


and radius 1, which arc call 4; and let Pp, q be the fluents of 
any other two successive terms, without the coefficients, the 
distance of @ from the first term a being n; then it is evi- 
dent that 9 == 2? p= 2",, and p= an—*,, Assume theref,. 

=br—ex’"—! ,/(a? —a?) ; then is q ora? p=) p —(2n— 1) 











2n—2 aoe exe x Eee bs _ (2a- 1 ea? 22-2, 
, ae z er 2) p PEST ah Tee 5 ei 
ar at Weer p —(2n—1)ea? P+(2n—1)ea? 


p ex? p = bp — “Sn — 1)ea? p+ 2nex?p, Then, comparing 
the coefficients of the like terms, we find 1 = en, and 6 = - 





1 2n—1 
(2n—1)ea? ; from which are obtained mo. and b= 7 a2. 
In— 2p — 72n—!1 Bes en8 a; os, 
Consequently a=! yaael wai Ya si , the general 


2n 
equation. between any two successive terms, and by means of 
which the series may be continued as far as we please. And 
hence, neglecting the coefficients, putting 4 = the first term, 


namely the arc whose sine is nf and 8, c, p, &c. the follow-. 
a, 


a2 A—2£4/ (0? — 2? 
ing terms, the series is as follows, a fe Gees (al) 


2 
35a? B— 2. 72) 2045 272) bt 
ee ete 2 ec. Now when 
“=O, this series ==0 ; and when z=a, the series becomes 1p 
| a? A 


=e 


PROMISCUOUS EXERCISES. 543 


Sa?B , 5a%e 








if — Sa a &e. ilies = 3:1416, or the series 
' 3 | 
is ip (1+ 40? + 5 “att 5 as &c.) 


So that, by taking, in A the coutiieate the general time of 
passing over any sincere DE will be 

w(e+l) | Qd-+-11. 
Ve Bopi te pX(I ayo? a0 —~ Sogo. 4° 
3 1 ata—x,/(a%—2?2)  2d+l 3B ds =). 
a 4al 2 ~ ' gadals” 

And hence, taking x==0, and doubling, the iinie of a whole 
vibration, or double ne time of pearing over cp will be equal 


a4 &e. —arc. sin. 











| we nid d+11.38 fe 4d?+-2dl-41a 
£0 aP/—9 x(q : Sen Q.4° '~ 428ds15~ 
1.3.5. 40d94+8d*l4+12dl?+5l2 1.3. ae 
Ea Gis 20480517 ere eg OF) 


wl 
Which, when a=O, or c=1, becomes only 1p Via? the same 


as in the last problem, as it ought. . 
Taking here the same oe as in the last spoblent 
viz, l/=3,a=1, w=2, w= 28800, g = 16,5; then 


29 2 
pW eee = °0040514, and the series is 1 -+- -006762 — 


"000175 + -000003, &c. = 1:006590; therefore 0040514 % 

By Song ; 
1°006590 = -0040965 = oaer the time of one whole vi- 
‘bration, and consequently 2451 vibrations are performed in a 
second ; which were 250 in the last problem. 


© : E 
PROBLEM 49. 


It is proposed to determine the Velocity, and the tume of Vibra- 
tion of a Fluid in the.arms of a Canal or woe Tube. 


Let the tube. ascper have its two 
branches ac, ce vertical, and the lower 
part cbe in any position whatever, the 
whole being of an uniform diameter or 
width throughout. Let water, or quick- 
silver, or any other fluid, be poured in, 
till it stand in equilibrio, at any hori- 
zontal line Br. Then let one surface be préded 0 or pushed 
down by shaking, from gs to c, and the other will ascend 
through the equal space ra; after which let them be oa 

mitte 





544 PROMISCUOUS, EXERCISES. 


mitted freely to return. The surfaces will then continually 
vibrate in equal times between ac and Ec. The velocity and 
times of which oscillations are therefore required. 

When the surfaces are any where out of a horizontal line, 
as at p and a, the parts of the ‘Auid in’ qpR, on each side, 
below ar, will balance each other; and the weight of the 
part in pr, which is equal to 2pr, gives motion to the whole. 
So that the weight of the part 2rpr is the motive force by 


of 2PF. 
which the whole fluid is urged, and thereforene > is —** is the 


accelerative force. Which weights being chests), to. 
their lengths, if J be the length of the whole fluid, or axis of 


the tube filled, and a = re or Be ; then is : the accelerative 
force. Putting theref. =r any variable distance, v the 


2a—~ 2a - 
velocity, and ¢ the time ; then pr =a — x, and #0 ” =f 





Matitedtc) fo, abel 
the accelerative force ; hence oy or 2gfs = 4 (ax —x2z) 3 


the fluent of which give v? = w (20x — x7), and v = 


2 
oa ei is the general expression for the Velocity 





J (4g X 


at any term. And when x = a, it becomes v = 2a 4/2 i 5 for 
the greatest velocity at 8 and F. 


Again, for the time, we have ; t ors 1Y/- 


the fluents of which give ¢ =1,/ — arc to versed sine = 
a 
and radius one, the general expression for the time. And 


l 
when « =a, it becomes t= 1p ¥ 5 for the time of moving 


l 
from ¢ to F, p being =3-1416, and consequently ip ae 


the time of a whole vibration from « to £, or from to a. 
And which therefore is the same, whatever 4B is, the whole 
length J remaining the same. 

And the time of vibration is also equal to the time of the. 
vibration of a pendulum whose length is 4J, or half the length 
of the axis of the fluid. So that, if the dength l be 78} inches, 
it will oscillate in 1 second. 

Scholium. This reciprocation of the water in the canal, is 

nearly 


PROMISCUOUS ‘EXERCISES. 546 


nearly similar to the motion of the waves of the sea, For, 
the time of vibration is the same, however short the branches 
are, provided the whole length be the same. So that when 
the ‘height i is small, in proportion to the length of the canal, 
the motion is similar to that of a wave, from the tep to the 
bottom or hollow, and from ‘the bottom to the top of the 
next wave ; being equal to two vibrations of the canal; the 
whole length of a wave, from top to top, being double the 
length of the canal. . Hence the wave will move forward by 
a space nearly equal to its breadth, in the time of two vibra- 
tions of a pendulum whose length is (41) half the length of 
the canal, or one-fourth of the breadth of a wave, or in the. 
time of one vibration of a pendulum whose length is the 
whole breadth of the wave, since the times of vibration are 
as the square roots of their lengths. Consequently, waves. 
whose breadth is equal to 39! inches, or 32% feet, will move 
over 32 feet in a second, or "1958 feet ina minute, or nearly 
© miles and a quarter in an hour. And the velocity ‘of greater 
or less waves will be increased or diminished i in the subdupli- 
cate ratio of their breadths. 

Thus, for instance, for a wave of 18 inches breadth, as 
A392 : 39h 3: 4/18 : of(39L X 18) = 21/313 = 26-5377 
the velocity of the wave of 18 inches breadth. 2 


PROBLEM 50. 


To determine the Time of emptying any Ditch, or sation 
&c. by. a Cut or Notch » from the Top to the Bottom Of it. 


Let « = az the variable height of water at 
any time : @ 
b = ac the breadth of the cut ; 


Bi 
d = the whole or first depth of water; _.— : a\. 
4 == the area of the surface of the water — 
in the ditch ; Aone 





. g = 16), feet. : | 
The velocity at any point p, is as ,/ Bp, thatis, as the ordi- 
nate pe of a parabola gEc, whose base is ac, and altitude az. 
Therefore the velocities at all the points in aR, are as all the 
ordinates of the parabola. Consequently, the quantity of 
water running through the cut apec, in any time, is t0 the 
quantity which would run through an equal aperture placed 
all at the bottom in the same time, as the area of the para- 
bola agc, to the area of the parallelogram ABGC, that is, ag 
2 to 3. 
But og :../%: ‘Pp : 2,/gx the velocity at ac ; atv 
2X2 vy ge Xba = 4ha f ge is the quantity discharged per. 
TWor: second” 


546 PROMISCUOUS EXERCISES. 
rhs 





second through asec ; and edascaaentiy’: —;-— is the ve- 
locity per second of the descending naiacee Fence then 
° , —3 
cil A agp —xiil": fini. 4 z the fluxion of the time of 
3A 4bx./ga 
descending. 


Now when a the surface of the water is constant, or the 
ditch is equally broad throughout, the correct fluent of this 
3A yt iat 


~——~—- for the general time of 
Bb e Jax 
sinking the surface to any depth x. And when « = 0, this 
expression is infinite ; which ah hy that the time of a com- 
plete exhaustion is infinite. 

Bat if d = 9 feet, b = 2 feet, a=21 X sh = 21000, 
and it be required ve exhaust the water down to ys of a foot 
deep; then x == ;, and the above expression becomes 


i x Sat = == 14400”, or just 4 hours for that time. 
4g 
And if it be required to depress it 8 feet or till 1 foot depth 
of water remain in the ditch, the time of sinking the water 
to that point will be 43’ 38”. 
Again, if the ditch be the same depth and length as before, 
but 20 feet broad bottom, and 22 at top; then the descend- 
ing surface will be a variable quantity, and, (by prob. 16 Prac. 


~X 20000 ; hence in this case the 





fluxion gives ¢ = 








Ex. on Forces), it will be 3 ; 


flux. of the time, or ————— , becomes —_ X 


aes hee SOAS UR 
correct fluent of witch i ist = >—>— X (—_ -— > 
B0./8.) fk Kd 


the time of sinking the water to any depth x. | 
Now when «= @, this expression for the complete ex- 
haustion becomes infinite. 
Butif .. x= 1 foot, the time ¢ is 42’ 56" 2 1, 
And when x= ;}; foot, the time is 3 50" 28"1, 


PROBLEM 51. 


Lo determine the Time of filling the Ditches of a Fortification 
6 Feet deep with water, through the Sluice of a Trunk of 3 
| Feet Square, the Bottom of which 1s level with the Bottom of 
the Ditch, and the Height of the supplying Water is 9 Feet 
above the Bottom of the Ditch. 


Let acpp represent the area of the vertical sluice, belle a 
square of 9 square feet, and az level with the bottom of the 
ditch 


PROMISCUOUS EXERCISES. 547 


ditch. And suppose the ditch filled to any height ar, the sur- 
face being then at er. . rave ! 
Put a = 9 the height of the head or supply ; ih 
b=3'= aB= Ac; | 


g ek 16 ‘ae Gi sue nes rit 
‘ =o Ts? 
A == the area of a horizontal section of Cp? 
the ditches ; ay) 
x=a— ar, the height of the head 
above EF. 





Then fg: ./x :: 2g : 2,/gx the velocity with which the 
water presses through the part aer» , and theref. 2 / gx x. 
AEFB = 26 J gz(a — x) is the quantity per second running 
through agers. Alse; che quantity running per second through 
HCODF is 24/gu X 11 ecpor = }) b,/ gx (b — a+<) nearly. 
For the real quantity is, by proceeding as in the last prob. 
the difference between two parab. segs. the alt. of the one 
being 2, its base 6, and the alt. of the other a’ — 6; and the 
medium of that dif. between its greatest state at an, where it 
is °;ap, and its least state at cp, where it is 0, is nearly 11 Ep. 
Consequently the sum of the two, or 2b,/gz(a + 11b — 2) 
is the quantity per second running in by the whole sluice 


a+1llb—zx 


acps. Hence then 3b4/ gx % cate =v 1s the rate 





or velocity per second with which the water rises in 


x —6a_ 2 
v 


ditches; andsov: —2::/:f=-< 








x 
b C—2x 
the. fluxion of the time of filling to any height az, putting 
e=a-+1lb. . 

Now when the ditches are of equal width throughout, a. 
is a constant.quantity, and in that case the correct fluent of 
6a JS Of Of Cm Sf Bo 
Saat ee aur re ree the ge 
neral expression for the time of filling to any height ax, or 
a— sax, not exceeding the height ac of the sluice. And 


this fluxion is t = 











ST 
by /ge og: 


) is the time of filling to cp the top of 


when x = ac =a — b = d suppose, then t = 


SEbJ/t fe—/fd 
Ve- fa era 
the sluice. 

Again, for filling to any height cn above the sluice, x de- 
noting as before a ~— ac the height of the head above au, 
 24/ gu will be the velocity of the water through the whole 
sluice ab: and therefore 2b? ,/ ga the quantity per secon, 

oan 





548 PROMISCUOUS EXERCISES. 


_, 2b? Ley a 
and MEP oy the rise per second of the water in the ditches ; 


ears: iar: ENS “2 oa 
consequently v: =a :: 1% 33 ; rab iy & 


general fluxion of the time; the correct fluent of which 
being 0 whens =a —b=d, Sila (fd — /x) the 


6? 
fime of filling from cp to ex. he 

Then tho sum of the two times, namely, that of filling 
from es cD, eae that of filling from cp to GH, 1s 
Pee a qosre, ~ log. ere Vive for the 

Fi Jie etd 
Hak sie sas And, using the numbers in the prob., 

//5— mee (et VIS 424/68 
this becomes aye yan Bf: /4atvo 
= 0°035772774, the ee in sf Hs of a the area of the 
length and breadth, or horizontal section of the ditches.. And 
if we suppose that that area to be 200000 square feet, the 
time required will be 7154”, or 15 59' 14”. 

And if the sides of the ditch slope a little, so as to be a 
little narrower at the bottom than at top, the process will be 
nearly the same, substituting for a its variable value, as in 
the preceding problem. And the time of filling will be very 
nearly the same as that above determined. 





PROBLEM 25. 


But if the Water; from which the Ditches are to be filled, be the 
Tide, which at Low Water is below the Bottom of the Trunk, 
and rises to 9 Feet above the Bottom of it by a regular Rise of 
One Foot in Half an Hour; it ts required to ascertain the 
Time of Filling it to 6 Feet high, as before in the last Problem. 


Let acps represent the sluice ; and when the tide has risen 
to any height cu, below cp the top of the sluice, without the 
ditches, let er Be the mean height of the water within. 
And put 6 = 3 = aB = ac; 


ele Ae wae . 
Aa = horizontal section of the ditches ; 
2% AG; 


Zz = AE. 
Then /g : ./EG :: 2g : 2/g lack a) the velo- 
city of the water Marne? ABFB ; and 
gt 4/bG i: 4g : 4,/ g(a—z) the mean vel. through ecHr : 
theref. 26z Vgls ws Ang ) is the quantity per sec. through aErs ; 
and 4b(a—2),/g(x— z) is the same through ecur ; 
conseq. 





PROMISCUOUS EXERCISES, 548 


conseq. 2b / g X (24 + 2) 4/ (x — z).is the whole through 
AGHB ies — This quantity divided by the surface a, 


ners evs x (2x +2) ,/ (x—z) = the velocity per second 


with which EF, or the surface of the water in the ditches, 

rises. Therefore. 

3A 
<eszi: lif hg 
1) a MY Sia, 
But, as cu rises uniformly 1 foot-in 36’ or 1800” there- 
fore 1: : 1800” : 1800r= ¢ werd pe of the tide rising 
Zz. 

sr * (2a-+2)./ (a — =z)’ 
or mz = (2a-+z),/(#—2z) . xis the fluxional equa. expressing 
as. 3200 


1200b/g 231 


through ac ; conseq. ¢ = 1800z = 


the relation between x and z; where m = 


or 13427 when a = 200000 square feet. 
Now ee ig the sar of this equation, assume z = 


‘cet dave es Poss vos So shall. 


do ad a2?+4p 2 a3+4asn+8c 10 
yf (e—z) =a ae rans 16 z” &¢ 


& 
2 


8 1] 
Qe z= 2n+ax?+aar*+cx * &c 











pi BU DR es. AO AB EF" 
OE ee craaaer er ae * x &e. 
2 La 
and mz = 5max pare MBX ~ 2 mex xt 'lmps ? x &c. 
Then equate the coefficients of ‘the like ternis, 
so shall and consequently 
4 
Sma * 2 A= Bn’ 
gms = 0, B= 0, 
11 ms Zaz = 24 
Page Hi rennn TYIT \, 
1AmD == — 24% = ZAB rah 
hig ees 875m” 
a&e.; &c. 


Which values of A, B, c, &c. substituted in the assumed value 
of z, give 
4 8 24 11 16 14 & 
a ee, : 
bm. 275m 875m4 
| pan 
or 2 = —-x* very nearly. 
pay se) : 


550 PROMISCUOUS EXERCISES. 


And when x = 3 = ac, then z = °886 of a foot, or {02 
inches, == ag, the height of the water in the ditches when 
the tide is at cn or 3 feet high without, or in the first hour 
and half of time. 
Again, to find the time, after the above, when { 
EF arrives at cp, or when the water in the ditches Gt erties 
arrives as high as the top of the sluice. ci iD 
The notation remaining as before, a FE 
then 2bz4/g(x—z) per sec. runs through AF, : 
and $b(3—z) 4/g(x—z) per sec. thro’ ep nearly; A B 


therefore 2b,/g X(12-+z) ./ (x — z).18 the whole per second 
through ap nearly. 


conseq. pee (12 + 2) s/f (a — 2) = vis the velocity per 


Maltod of the point £ ; ms therefore, 
a BA 

| » 2bi/g “Cee \4/(4—z) 
mz=(12-+-z) 1 (x—z) . x, where m=—— 





vizier st = = 1800 x, or 


ee =237, nearly. 
Assume z Sax! -+ne? + cx? -+-pxt &c. So shall 
a2 ‘s2+4—p 3 ,3+4ap+8c 4 


/ (x—z) ae ~~ alia z pes tie 8 4 ie &C. ; 


12+2=12-bar? tart ton! &c. ; 
(12-+-z). /\(—-7): saiects— ena! (342 -+-6n) a2 Xe. ; 


mz=imaxtet4 mBx? 2+ Emon? a &c. 
Then, equating ‘the like terms, &c. we have’ 








Sag es 24 Ee 8 8S OLN 
A ze B= — me C= Bra D om nearly, &c. 
H eat aes & 
i$ e =—x~ tiki 2 ——L ca re x? ° 
ence Z=— x — oi ae rg, 12 nis {3 &e 
8 3 , 
Oriz= ee 


But by the first process, when +=3, z=:886 ; which sub- 
stitated for them, we have 2 = °886, ana the series = es 463 - 
therefore the correct fluents are 


‘ ‘ 
e—886= ~ 1-634 Oe 24 otic: 
FTL) m? 


: 
or 2+" 714 a? _ txt &e. 


And when z = 3 = ac, it eee x = 6-369 for the height of 
the tide without, when the ditches are filled to the top of the 
sluice, or 3 feet high ; which answers to 3h 11’ 4”, 

Lastly, 


PROMISCUOUS EXERCISES. 551 


Lastly, to find the time of rising the remaining 3 feet above 


ihe top of the sluice; let — 
x=ce the height of the tide above cD, hid ur sewy HL 
z=ceE ditto in the ditches above cp ; aed 


and the other dimensions as before. 
Then / g : /nc::2g:24/g(x—2) =the _ 
velocity with which the water runs throughthe 

whole sluice ap ; censeq. aDX24/g (4-2) = A B 
18,/ g (x — 2) is the quantity per second running through the 


sluice, and aed YW (x—z) = the velocity of z, or the rise 


of the water in the oe per second; hencev : 2:: 1": 7 


Por esate 
= - = =1800z, and mz = x of (x—z)* is 
v 18g” v Yess ) a 

the fluxional equation ; where m= Toe Ene 9 


To find the fluent, 
3 4 5 6 
Assume 2 = az?-+-pax?-++cx?-+- pa? &c. 
3 4 § 
Then x—z=a — ax? —pa?—ca? &c. 
; +. Adz, a?--4p 2¢ 
x 4f(a—2) =a" x == 4 5 enous ot x x &e. 
2 8 
mz =2nax?s +4npeta + §nce?: x &e. 
Then equating the like terms gives 
aD 1 —1 
Si Ga” 90m 9 BION 
i 
2 ye x3 
sai  S0n? sone didnt. 
But, by the second case, when.z = 0, « = 3-369, which 
being ‘used in the series, 1 us is i‘ 936 ; therefore the correct 





A 


9 3 
Hence z = —x? — 
3n 


1 
uentis 2 = — 1-936 2 5 a &c. And when 


z =: 3,z 7; the heights above the top of the sluice ; an- 
swering to 6 and 10 feet above the bottom of the ditches. 
That is, for the water to rise to the height of 6 feet within 
the ditches, it is necessary for the tide to rise to 10 feet with- 
out, which just answers to 5 hours ; and so long it would take 
to fll the ditches 6 feet deep with water, their horizontal 
area being 200000 square feet. 

Further, when x = 6, then z = 2-117 the height above 


— 





* Note. The fluxional rea mi ma 2 4/(0~z) may be integrated without 
serieg,-E DITOR, 
the 


552 PROMISCUOUS EXERCISES. 


Gs 


the top of the sluice ; to which add 3, the height of the 
sluice, and the sum 5:117, is the depth of water in the ditches 
in 4 hours and a half, or when the tide has risen to sa nO oe 
of 9 feet without the ditches. 

Note. In the foregeing problems, concerning the afflux wf | 
water, it is taken for granted that the velocity is the same as 
that which is due to the whole height of the surface of the 
supplying water : a supposition which agrees with the prin- 
ciples of the greater number of authors : though some make 
the velocity to be that which is due to the half height only : 
and others make it still less. 

Also in some places, where the difference between two pa- 
rabolic segments was to be taken, in estimating the mean ve- 
locity of the water through a variable orifice, 1 have used a 
near mean value of the expression ; which makes the _opera- 
tion of finding the fluents much more easy, and is at the same 
time sufficiently exact for the purpose in hand. 

We may further add a remark here concerning the method © 
of finding the fluents of the three fluxional forms that occur 
in the solution of this problem, viz. the three forms mz = 
Cane Vv (@ — 2) x, and mz = (12+ 2) f (x#-2) z, and . 
mz = Vf (r— —2) xx the fiuents of which are found by assuming 
the fluent mz in an infinite series ascending in terms of x with 
indeterminate coefficients a, B, c, &c. which coefficients are 
afterwards determined in the usual way, by equating the cor- 
responding terms of two similar and equal series, the one se- © 
ries denoting one side of the fluxional equation, and the other 
series the other side. Ly similar series, is meant when they 
have equal or like exponents ; though it is not necessary that 
the exponents of all the terms should. be like or pairs, but on- 
ly some of them, as those that are not in pairs will be can- 
celled or expelled by making their coefficients = 0 or nothing. 
Now the general way to make the two series similar, is to 
assume the fluent z equal to a series in terms of x, an as- 
‘ nine or descending, as here 

2 = a ats +7425 &c. for ascending, 
or 2 = 2 +a7—-+a2"" 2° &e. for a a descending 
series, having the exponents r, rs, r + Qs, &c. in arith- 
metical progression, the first term 7, and common difference 
$3 without the general coefiicients a, B,c, &c. till the values | 
of the exponents be determined. In terms of this assumed 
series for z, find the values of the two sides of the given 
fluxional equation, by substituting in it the said series instead 
of z ; then put the exponent of the first term of the one side 
equal that of the other, which will give the value of the first 
exponent 7; In like manner put the exponents of the two 2d __ 
terms 


PROMISCUOUS EXERCISES. 953 


terms equal, which will give the value of the common differ- 
ence s; and hence the whole series of exponents r, r+ s, r 
zh 2s &c. becomes known. 

Thus, for the last of the three fluxional equations above 
mentioned, viz. mz=4/ (e—2)x, or only z=3 Vf (4—2) x5 
having assumed as above z ==27-+ a’!s &c. and taking the 
fluxion, then z=27~-! x + 27*s-!y + &c. omitting the co- 
efficients; and the other arte of ne equation 4/ (w--z) 4 = 


af (% =a” —2"* &c.) = aig — a2 Ke. Nee the exponents 
% the first terms made equal, give r—1= 4, theref. r=1-+4 
one ; and those of the 2d terms made aa give r-+s— l=? 
4, theref. s—1=—1, and s=1—1=1: conseq. the wos 
sean series of exponents r, r-+s, r-+2s, &c. become 3, 4, 3, 
&c. as assumed above. 
Again, for the 2d equation mzorz= (12 + 2) ./(x-z)x 
= (a+ z) / (% — z)x; assuming 2=2"--2"" &c. as before, 


then == a7—1 7 + a7ts—1 yx Ke. and / (% —2) ca? x —x"Fx 
&c. Bosh as above ; this mult. by a . zora+2’ +27 &c. 


gives ax? 2 an} xz &c: then equating the first exponents 
gives r—1=} or r=, and r Bai gti or s= 1—1=1; 
hence the series of exponents is 3, 4, &, &c. the same as the 
former, and as assumed above. 

Lastly, assuming the same form of series for z and z as in 
the above two cases, for the 1st fluxional equation also, viz. 
mz =(2x-bz),/ (a —2)z: then yf (e—z)z =2%g — 272 Ke, 
which mult. by 27-+-z, gives 2x2 sir" &c.: here equat- 
ing the first exponents gives r—1 = 2 or r = $, and  Seglantanly 
the 2d exponents gives r -+-s—1 = r +1, or := =2,; hence 
the series of exponents in this case is 4, £, 1, &c. as used 
for this case above. Then, in every case, the general co- 
efficients a, 8B, c, &c. are joined te the assumed terms a’, x7**, 
&c. and the whole process conducted as in the three series 
just referred to. 

Such then is the regular and legitimate way of hieesdiig 
to obtain the form of the series with respect to the exponents 
of the terms. But in many cases we may perceive at sight, 
without that formal process, what the law of the exponents 
will be, as I indeed did in the solutions in the series above 
referred to ; and any person witha little practice may easily 
do the same. 


PROBLEM 
Vor. Il, "1 


554 PROMISCUOUS EXERCISES, 


PROBLEM 53. 
To determine the Fall of the Water in the Arches of a Bridge. 


The effects of obstacles placed in a current of water, such 
as the piers of a bridge, are, a sudden steep descent, and an 
increase of velocity in the stream of water, just under the 
arches, more or less in proportion to the quantity of the ob- 
struction and velocity of the current ; being very small and 
hardly perceptible where the arches are large and the piers 
few or small, but in a high and extraordinary degree at Lon- 
don-bridge, and some others, where the piers and the ster- 
lings are so very large, in proportion to the arches. This is 
the case, not only in such streams as run always the same 
way, but in tide rivers also, both upward and downward, but 
much less in the former than in the latter. During the time 
of flood, when the tide is flowing upward, the rise of the 
water is against the under side of the piers ; but the differ- 
ence between the two sides gradually diminishes as the tide 
flows less rapidly towards the conclusion of the flood. When 
this has attained its fall height, and there is no longer any 
current, but a stillness prevails in the water for a short time, 
the surface assumes an equal level, both above and below 
bridge. But as soon as the tide begins to ebb or return 
again, the resistance of the piers against the stream, and the 
contraction of the waterway, cause a rise of the surface above 
and under the arches, with a full and a more rapid descent in 
the contracted stream just below. The quantity of this rise, 
and of the consequent velocity below, keep both gradually in- 
creasing, as the tide continues ebbing, till at quite low water, 
when the stream or natural current being the quickest, the 
fall under the arches is the greatest. And itis the quantity 
of this fall which it is the object of this problem to deter- 
mine. 

- Now, the motion of free running water is the consequence 
ef and produced by the force of gravity, as well as that of any 
other falling body. Hence the height due to the velocity, that 
is, the height to be freely fallea by any body to acquire the 
observed velocity of the natural stream, in the river a little 
way above bridge, becomes known. From the same velocity 
also will be found that of the increased current in the nar- 
rowed way of the arches, by taking it in the reciprocal pro- 
portion of the breadth of the river above, to the contracted 
way in the arches ; viz. by saying, as the latter is to the for-_ 
mer, so is the first velocity, or slower motion, to the quicker. 
Next, from this last velocity, will be found the height due to 

it 


PROMISCUOUS EXERCISES. . 555 


it as before, that is, the height to be freely fallen through by 
gravity, to produce it. Then the difference of these two 
heights, thus freely fallen by gravity, to produce the two 
velocities, is the required quantity of the waterfall in the 
arches ; allowing however, in the calculation for the centrac- 
_ tion, in the narrowed passage, at the rate as observed by Sir 

I. Newton, in prop. 36 of the 2d book of the Principia, er by 
other authors, being nearly in the ratio of 25 to 21. Such 
then are the elements and principles on which the solution 
of the problem is easily made out as follows. 


Let 6 = the breadth of the channel in feet ; 
v = mean velocity of the water in feet per second; * 
c = breadth of the waterway between the obstacles. 


QI ! 
Now 25: 21::c: 95° the waterway contracted as above. 


21 25 ee ; 
And on" a a =, the velocity in the contracted way. 
Also 32? : v® :/ 16: -,v?, height fallen to gain the velocity v. 
256 25b 25b 
And 322 :(——v)2 :: 16: (——)?2x p42, di 0. 
n Cm” 16 ror, X grv?, ditto for the vel ae 


Q 2 2 
Then (20) 9x pag - is the measure of the fall required. 


4 
25d Oo. ied Ha 
a) —1] X= is a rule for computing the fall. . 


Or rather 


Or [( 
1-42b2 —c? 
64c? 








Xx? very nearly, for the fall. 


Exam. 1. For London-Bridge. 


By the observations made by Mr. Labelye in 1746, 
The breadih of the Thames at London-bridge is 926 feet ; 
The sum of the waterways at the time of low-water is 236 ft.; 
Mean velocity of the stream just above bridge is 31 feet per 
second. But under almost all the arches are driven into the 
bed great numbers of what are called dripshot piles, to pre- 
vent the bed from being washed away by the fall. These 
dripshot piles still further contract the waterways, at least 2 
of their measured breadth, or near 39 feet in the whole ; so 
that the waterway will be reduced to 197 feet, or in round 
numbers suppese 200 feet. 

Then 6 = 926, ¢ = 200, v = 31: 
Hence 


536. PROMISCUOUS EXERCISES. 


1-42b2—c2_ 1217616 —40000 
Hence 


— —_—= “46. 
~ 64¢2° 64 x 40000 
192 
And v? = —-=10 ye. 


Theref. 46 X 103, = 4:73 ft. 83 in. the fall required. By 
the most exact observations made ‘about the year 1736, the 
measure of the fall was 4 feet 9 inches. 


Exam. 2. For Westminster-bridge. 


Though the breadth of the river at Westminster-bridge is 
1220 feet ; yet, at the time of the greatest fall, there is water 
through only the 13 large arches, which amount to but 820 
feet ; to which adding the breadth of the 12 intermediate 
piers, equal to 174 feet, gives 994 for the breadth of the ri- 
ver at that time ; and the velocity of the water a little above 
the bridge, from many experiments, is not more than 2} feet 
per second. 

Here then 6 = 994, ¢ = 820, v = 21 = 2. 

1:4262—~c? _ 1403011— 672400 


Eee Ot Pe os = . 9 e 
ate Pe 64 x 672400 A 
812 
And v? = 6 = 535 
Theref. -01722 x 6. = 0872 ft. =1 in. the fall requir- 


ed ; which is about half an inch more than the greatest fall 
observed by Mr. Labelye. 


And, for Blackfriar’s-bridge, the fall will be much the same 
as that of Westminster. 


ADDITIONS, 


Neouereyp 


ADDITIONS. 


BY THE EDITOR, R. ADRAIN. — 


Vew Method of determining the Angle contained by the chord 
a two sides of a Spherical Triangle. 
) See prop. v. page 77, He 2. 


‘THEOREM. 


If any two sides of a Spherical Triangle be produced till the 
continuation of each side be half the supplement of that 
side, the arc of a great Circle joining the extremities of the 
sides thus produced will be the measure of the angle con- 
tained by the chords of those two sides. 


’ DEMONSTRATION. 


Let the two sides as, ac of the spherical 
triangle asc be produced till they meet in 
G, and let the supplements Be, ca, be bisect- 
ed in p and gs, also let the chords ams, anc 
of the arcs as, ac be drawn ; and the great 
circular arc pre will be the measure of the 
rectilineal angle contained by the chords ~ 
AMB, ANC. 





AN 

Let the diameter ac be the common section of the planes 
of ana, acc, and Fr the centre of the sphere, from which draw - 
the straight lines Fp, FE. 

Since, by hypothesis, cr is the half of cc, therefore the 
angle at the centre Gre is equal to the angle at the circumfe- 
rence Ganc (theo. 49, Geom.) and therefore anc and Fu, be- 
ing in the same plane, are parallel: in lke manner, it is 
shown that rp and ams are parallel, and therefore the recti- 
lineal angles Bac and pre, are equal, and consequently, since 
pe is the measure of the angle pre, it is also the measure of 
the angle contained by the chords ams and anc. Q. ED. 


New 


556 ADDITIONS. 


New method of determining the Oscillations of a Variable 
Pendulum. 


The principles adopted by Dr. Hutton in the solution of 
his 45th problem, page 537, vol. 2, are, ic my opinion, erro- 
neous. He supposes the number of vibrations made in a 
given particle of time to depend on the length of the pendu- 
lum only, without considering the accelerative tension of the 
thread ; so that by his formula we have «x fiuite number of vi- 
brations performed in a finite time by the descending weight, 
even when the ascending weight is infinitely small or nothing. 
Besides, the stating by which he finds the fluxion of the num- 
ber of vibrations, is referred to no geometrical or mechanical 
principle, and appears to be nothing but a mere hypothesis. 
The following 1s a specimen of the method by which such 
problems may be solved according to acknowledged princi- 
ples. 


PROBLEM. 


If two unequal weights m and m' connected by a thread passing 
Jreely over a pulley, are suspended vertically, and exposed to 
the actron of common gravity, at 8 required to unvestigate the 
number of vibrations made in a given time hy the greater 
weight m, supposing at to descend from the point of suspen- 
sion, and to make indefinitely small removals from the verti- 
cal. 

SOLUTION. 


Let the summit a of a vertical ascnr be the 
point from which m descends, 8 any point in Ae 
taken as the beginning of the plane curve smpn 
described by m, which is connected with m’ by 
the thread am. Let. mc be at right angles to ag, 
and put ac=2, cm=y, am=r ; also let t,t and tT 
be the times of the descent of m through the ver- 
tical spaces as, ac, and Bc ; g=32} feet, = the 
measure of accelerative gravity ; f= the mea- 
sure of the retarding force which the tension of rn 'R 
the thread exerts on m in the direction ma, and c = the in- 
definitely small horizontal velocity of m at B. 


bab b> 


1 OT ee age Ln the vertical action of the tension on m; 


fe 


and theref. he opie the true accelerative force with which 


m is urged in a vertical direction. 
Again, 


ADDITIONS. 550 


Again, rye hfs SY the horizontal action on m produced 
yt \ 


by the tension of the thread am. Thus the whole accelerative 
forces by which m is urged in directions parallel to-a and y, 
are g- Jt and oe the former of these forces tending to in- 
r 
crease x, and the latter to diminish y; and therefore by the: 
general and well known theorem of variable motions (See 
Mec. Cel. B. 1, Chap. 2), we have the two equations 
gl OE PE A MEIEY 
FC 2 z and me e ° 
But by hypothesis, the angle mac is indefinitely small, we have 
Qm!é ig : 
therefore = =1,and Nag saa, = agiven quantity ; our first 


fluxional equation therefore becomes 
x 
ag. 8 weed 


of which the proper fluent is x==1(¢—/) ¢2: and by substitut- 
ing for x the value just found, our second fluxional equation 
becomes | 

Bit Vt by me lap a Ae 
- z-j # OK -+py = 0, (putting p = ay A aaa 
Now when p is less than }, let gy=,/1 —p, and in this case the 





iy Bays i d 
correct fluent of the equation —e bry=0, is easily found tobe 


ee Cg ay pee 
PT TE sn ah 
cae 36 (>) 


from which equation it is manifest that as ¢ increases y also in- 
creases, so that m never returns to the vertical, and there are 
no vibrations. Again, when p =1, the correct fluent of the 
same fluxional equation is 


y aca t 
ow & hyp. log. (=). 
So that in this case also, when t increases y increases, andthe 
body m never returns to the vertical. Since in this case p= 
Aa : 
til = =1, therefore 17m'=m, and therefore by this case 
m—m 4 
and the preceding, there are no vibrations performed by the 
descending weight m when it is equal to or greater than 17 
times the ascending weight m’. | 
But when pis greater than }, put n = 4/p — }, and in this 





4 
case the correct equation of the fluents is 
¥y 


G 


eet 
a 


560 ADDITIONS. 


et 
es 


j ~: 


This pagAHO shows us that we shall have y=0 as often as 





n. hyp. log. - - ’ becomes equal te any complete number of se- 


Be arin farcicesk : if Ftherefare w= 3:1416, and n = any num- 

ber in the series 1, 2, 3, 4, 5, &c. we can have y=0 only when 
t ‘ 

n. hyp. log. <=NR, from which we have bce, <, suppos* 

ing hyp. log. e=1, and therefore 


eed 


. which shows the relation between the ee of vinpubiohe N 
and the time 1 in which they are performed. 
Hence it is manifest, that the times or durations of the se- 
veral successive vibrations constitute a series in geometrical 
progression, 


PESCRIPTIVE 


[561] 


DESCRIPTIVE GEOMETRY. 





CHAPTER I. 


Containing the First Principles of Descriptive Geometry, 
with Illustrations. 


Descriptive Gromerry is the art of determining by con- 
structions performed on one plane the various points of lines 
and surfaces which are in different planes, The principle on 
which this art is founded, consists in projecting the points of 
any line or surface on two given planes at right angles to each 
other. These two planes are usually denominated the hori- 
zontal and vertical planes, or the fundamental or primitive 
planes, or the planes of projection. In the constructions the 
vertical plane is supposed to have revolved about the line of 
their common intersection, and to be coincident with the ho- 
rizontal plane ; and it is by means of this coincidence that 
both the projections on the horizontal and vertical planes are 
effected by constructions performed on the horizontal plane, 


D C 





A B 
To illustrate this, let ascp be the horizontal plane, and 


eron the vertical plane at right angles to it, and meeting it in 
Vou. I. 72 the 


568 DESCRIPTIVE GEOMETRY. 


the straight line of their common section er. Suppose P te 
be any point in space, from which on the plane azcp let fall the 
perpendicular rr’, meeting that plane in P’ ; and on the vertical 
ban EFen, let fall from Pp the perpendicular pp” weeting it in 
; then r’and P” are the projections of the point pr on the 
Bniaitive planes : and it is obvious that the projections of any 
other point besides p cannot be coincident with both the points 
p’ and P”’, and therefore, when the points P’ and P” are given, 
there is but one point p of which they are the projections. 

F rom the point-p’ draw in the horizontal plane the ‘Straight 
line p’k at right angles to the common section er, and join PK. 
It is obvious that r’x is at right angles to the plane EFGH, and 
by supposition pp” is at right angles to the same piane, conse- 
quently pe”, p’x are parallels, and therefore in one plane ; and 
since the angle rr’« is a right angle, therefore ppp” is also a 
right angle : and because pr’x is a right angle, it follows that 
px’ is likewise aright angle ; thus it appears that the plane 
figure Pp KP’ is a rectangle, and the two distances r’k, P’K, are 
equal to the two projecting perpendiculars PP’ ‘and pr’. Those 
perpendiculars pr’’ and vr’, or their equals r’x, rx, are called 
the ordinates of the point rp. — 

Suppose now, after the points pr’ and P” are determined, that , 
the plane ercH revolves about its intersection er from its po- 
sition at right angles to ancp, until it coincides with the horizon- 





AL B 


tal plane : during this revolution the straight line xp’ in its 
motion continues at right angles to the common section EF ; 
and when the vertical plane ercu, has coincided with the 


horizontal 


DESCRIPTIVE GEOMETRY. 563 


horizontal plane azcp, the line p’ of the former plane evident- 
ly falls in the continuation of r’k ; sothat r’x, ke”, make one 
straight line at right angles to er, and lying in the horizontal 
plane ; the distances r’k, px, being the ordinates, or co-ordi- 
nates to p, the point in space. 

The point p in space is said to be given. when the two 
perpendiculars. or ordinates, PK and P’K, are given in magni- 
tude and position ; and a point sought, P is said to be found 
when the two ordinates pk and p’K have been found. The 
various positions of the projections P’ and p” corresponding 
to the different situations of the point vr in space, should be 
clearly conceived by the learner ; on this account the follow- 
ing varieties of position deserve attention : and itis particu- 
larly to be noted, that the horizontal projection of the point 
in space is marked with one accent, and the vertical projec- 
tion with two accents, by means of ‘which the several points 
of the horizontal and vertical planes will be samy distin- 
guished. : | 





If the point P which is to be projected, be in the ground 
line or common intersection eF of the fundamental planes, its 
projections P’, p’, must evidently ceincide with the point itself 
as in N°. 1. 

ifthe point ep be in one of the fandamental planes bat not 
in the other, let it first be in the horizontal plane at Pr’, as in 
N*.2, N°. 3. | In each of which the vertical projection PY 


falls on the ground fine: im N®.2, the point. r’ is before 
the 


864 DESCRIPTIVE GEOMETRY. 


the vertical plane, and in N°. 3. the point rv’ is behind thé 
vertical plane. Next let the point p be in the vertical but: 
not in the horizontal plane, as at re’ N°. 4, No. 5. In each 
of these cases the horizontal projection is manifestly on the 
ground line at re’. In N°. 4, the point p orp” is in the verti- 
cal plane directly above the point r’ of the ground line, and 
by the revolution of the vertical plane zrcu into a horizontal 
position, the point r” falls behind the ground line er. In N°. 
5, the point p or p”is directly below the point P’ of the ground 
line in the continuation of the vertical plane ercu below the 
horizontal plane ancy ; and by the same revolution of eran as 
before, the point vr“ of the vertical plane immediately below Pp’, 
is brought up to the horizontal plane ; so that in this Jast case 
the point P” is before the ground line: and therefore the 
points ep” and e” of N°. 4 and N°. 5, fall on opposite sides of 
eF on the horizontal plane by the revolution of the vertical 
plane. When the point of space P is in neither of the primi- 
tive planes, there are four different situations in which it may 
be found, that require to be particularly distinguished from 
one another. 


AN Rea pee intsrinahehenbeen: 





i. When the point p is above the horizontal plane and bes 
fare the vertical plane. In this case the horizontal projection 
r’ falls before the ground line zr, and the vertical projection 
Y falls behind er ; the horizontal and vertical ordinates be- 
ing KP’ and xP”. To conceive distinctly the place of the 
point Pp, take in ey the distance KL equal to xe”; and on 
the horizontal plane complete the rectangle xumr’. Ima- 
gine now that the rectangle ximr’ revolves about its fixed 
side kr from a horizontal to a vertical position by the ascent 
ef the rectangle above the horizontal plane ; and when the 

rectangle 


DESCRIPTIVE GEOMETRY. 565. 


tectangle Kump’ is in this vertical position, its angular point M 
will coincide with the point pe of which the projections are vr’ 
and e”: and the angular point 1, after having described a 
quadrant of a circle on the primitive vertical plane, will coin- 
cide with that point of it which is the vertical projection of r, 
and which is denoted by the point p”. 

2. When the point P is above the horizontal plane and be- 
hind the vertical plane. In this case the projections P and 
Pe" both fall behind the ground line er, in the same straight 
line xp’p’, Having made x1 equal to xp”, and completed the 
rectangle, suppose it to revolve about its side xp’ which re- 
main fixed by ascending from a horizontal to a vertical posi- 
tion, and the point will coincide with the point ry, which ts 
conceived to be directly above P’, and at an altitude equal te 
KP” or P’M. 


D sa hg ACORN 8: 






A. B 


3. When the point r is below the horizontal plane and behind 
the vertical plane. In this case the point r’ of the horizontal 
projection of p falls behind the ground line er, and the vertical 
projection p’ falls before it. To determine the situation of 
the point rp, corresponding to the projections Pp’ and P’, com- 
plete the rectangle gimp’ as before ; and suppose it to revolve 
about the fixed side p’x, from a horizontal to a vertical posi- 
tion by the descent of the side im, so that the point m may 
be directly below the point rv’ of the horizontal plane. Then 
will the point m coincide with the point P, of which the hori- 
zontal and vertical projections are p’ and Pp”. When the rect- 
angle KiMp’ is in the vertical position, and m coinciding with 
p, mP’is the projecting line or ordinate upwards from P on the 
horizontal plane, and mr is the projecting line from r on the 
vertical plane ; the point x being in the vertical plane direct- 


ly 


566 DESCRIPTIVE GEOMETRY. 
D wiih C 








B 


ly below x. The point x immediately below x is that which is 
denoted by the point P”; it is brought into the point pe” coin- 
ciding with the horizontal plane by the revolution of the ver- 
tical plane about Er, from a vertical to a horizontal position, 
the upper part of the vertical plane falling behind er toward 
cp, and the lower part rising so as to coincide with the hori- 
zontal plane before er towards as. 

4, When the point r is below the horizontal plane and be- 
fore the vertical plane. In this case both points of projection r’ 
and pr’ are before the groundline er. The exact situation of F 


D C 





, A B | 
corresponding to P’ and Pp”, may be known by constructing the - 
rectangle kim’ as before, and conceiving it to revolve about 
xp’, by descending from a horizontal to a vertical position. In 
this vertical situation of the rectangle, the point m coincides 

with 


DESCRIPTIVE GEOMETRY. 567 


with the point rp, whichis directly below p’, and mr’, mn are 
the two projecting lines by which P is represented at P’ on 
the horizontal plane, and at i directly below x in the verti- 
cal plane. By the revolution of the vertical plane about xr, 
the point of projection t directly below « is brought upwards 
into the point Pp” on the horizontal plane. . 

As a farther elucidation of the general principle, let us con- 
sider the projections of straight lines. : 


pS fa ek one 





A sor | 5 


Let ascp be the horizontal plane, and eren the vertical 
plane at right angles to the former, and meeting it in their com- 
mon intersection EF; and let rq be any straight line in space. 
From Pp and e anytwo points of the straight line Pe imagine two 
straight lines pp and a@ to be drawn at right angles to the hori- 
zontal plane apcp and meeting it in P and @’; and from the same 
points pand q two other straight lines pr” and qq” to be drawn 
at right angles to the vertical plane ercn, meeting it in the 
points Pe” and q”. Draw p’k, @'t at right angles to the ground 
line gF ; join P’K, Q“L 3 and we have as before the rectangle 
PP’KP’, of which the sides pr” and pr’, or their equals p’x and 
r’K, are the ordinates of the point r : and in like manner ae” 
and ae’, ov their equals 9’t @1, are the ordinates of the point 
of space a. | 

Suppose now a plane to pass through the line in space ra, 
and either of the perpendiculars pr’, and q@ : and it is easy 
to perceive that it will pass through the other perpendicular, 
and meet the horizontal plane in the straight r’a’ which joins 
the points P’ and’, It is also evident that all the perpendi- 

culars 


568 DESCRIPTIVE GEOMETRY. 


culars let fall on the horizontal plane from the several points 
of the line in rq will meet the horizontal plane in the straight 
line p’q’; the straight line e@’ is therefore called the hori- 
zontal iene of the straight line ra. From this construc- 
tion it is plain that the projection of a straight line on a plane 
is a straight line on the plane passing through the projections 
on the same plane of any two points of the proposed straight 
line ; or which amounts to the same thing, the projection of a 
straight line on a plane is the common intersection of this plane 
and another plane at right angles to the former, and passing 
through the straight line. 

From this definition it is manifest that p’g” is the projection 
of pa on the vertical plane ; so that P’ Q’ and p’9” are the ho- 
rizontal and vertical projections of the straight line Pe. 

Conceive now that after the projections of Pra are thus 
inade, the vertical plane ercH revolves. about the common 
section er from a vertical position till it coincide with the ho- 
rizontal plane, the higher part of the vertical plane being 
supposed to fall behind the common section Ex ; ; the straight 


ID .. : Cc 





lines which are at right angles to EF will fallin the continuation 
of p’k and @’L: so that the projections of re will now obtain 
the positions Pa, P’a” on the same plane ; the ordinates p’x, KP” 
making one straight line, and @‘L, La ’ also melig a one straight 

line. 
The various positions of the projections P‘q’, ’q” will be 
fully exemplified in the subsequent problems ; a is sf oecie 
"te 


DESCRIPTIVE GEOMETRY. 569 


to observe here that a straight line ee is said to be given in 
space when its projections PQ’ and Pp” @’ are given; anda 
straight line re is said to be found when its projections rq’, 
p’@” are found. To which we may add that the two planes 
passing through pq and each of the projections p’q’ and p’9” 
are called the projecting planes of rq ; of course a straight 
line will always be given in position, when we have the inter- 
section of its projecting planes. 

When a plane exists in space it is referred to the planes of 
projection by means of its two intersections with those two 
planes. Let ascp and ercH be the horizontal and vertical 
planes ; and let xq’rQ” be any other plane : this plane will in 
general cut both the planes of projection; the horizontal 
plane in the straight line xe’ and the vertical plane in the 
straight line rq”. These intersections xe’, Ke” are called 
the traces of the plane xe’ nq”; the former x’ being the 
horizontal trace of the plane, and xq” its vertical trace. 
When the vertical plane ercH revolves about the ground 
line from a vertical to a horizontal position, the vertical, 
trace xe’ will be in the horizontal plane, and the two 
traces will then be in the same horizontal plane, meeting each 
other in the point x in which the plane xe’R@” cuts the 
ground line er. A plane is said to be given in position when 
its horizontal and vertical traces are given. 


D | REE RY 











B 


The various positions of the traces of a plane according to 
the situation of the plane will be exhibited in some of the 


problems in the following chapter. 
Vou. I, 73 CHAPTER 


6054 


CHAPTER Il. 
Containing F hee eticl Problems. 
PROBLEM I. 


If a straight line be. given by its projections, at as required to 
find its height above the horizontal plane at any povnt of its 
projection on that plane. 


Let aecp be the hori- pH 
zontal plane, and eFcp 
the vertical plane, which 
- by revolution about their 
- common intersection or, 
ground line er, is brought 
into a horizontal position. 0 

Let p’e and p”@” be the 
two projections of the 
given line, the former ra’ 
on the horizontal plane, 
the latter p’e” on the ver- 
tical plane ; itis required A B 
to find the altitude of the given line above any point P' of the 
horizontal projection P'a’. 

Draw rx at right angles to the ground line er, and produce 
rk if necessary to meet p’Q” the vertical projection in P” : 
and xe” will be the height of the given straight line above the 
point P’. Gass 

Because P’@’ is the horizontal projection of a straight line, 
therefore the point rp’ is the horizontal projection of some 
point p of that line; but. the two projections of a point are 
always in the same straight line at right angles to the ground 
line, therefore the vertical projection of p is in PKP’: and 
because PQ” is the vertical projection of the given line, the 
vertical projection of p must be in P’@”, therefore the verti- 
cal projection of P is in the point P’ which is the intersection 
of pxr’ and p’q”. Therefore p’x, Px are the horizontal and 
vertical ordinates of the point p, and xp“ is equal to the height 
of the given line above the point pr’. | 

, If ge’ be any other point in the horizontal projection PQ’ ; 
draw as before 9/L@” at right angles to er, and 1q” will be the 
height required. In this second. case the point q” falls before 


the 


C 





DESCRIPTIVE GEOMETRY. B74 


the ground line; and therefore, agreeably to the illustrations of 
the projections of a point given in the first chapter, the dis- 
tance Lq’ is a depresssion below the horizontal plane : that is, 
the point a, of which Q’ and Q” are the horizontal and vertical 
projections, is directly below the point q’ of the horizontal 
plane, its distance below Q being equal to rq”. Thus it ap- 
pears that the point of the given line, of which p and p’ are 
the projections, is above the horizontal plane and before the 
vertical plane, but that the point of this line of which @’ and 
q” are the projections, is below the herizontal and behind the 
vertical plane. 


Again, let P’ a’, and P” 
Q”be the horizontal and 
vertical projections of a 
straight line, and ap the 
ground line. In this figure 
the point of the given 
line q, of which @’ and 
are the projections, Is a- 
bove the point q’ of the 
horizontal plane, at a 
height equal to the ordi- 
nate pq’; and the point 
of the line p, of which p’ and p” are the projections, is s below 
the point P’ of the horizontal plane. the depression of the 
point being equal to the ordinate xe’: whence it appears that 
the point of the given line projected into @ and @’ is above 
the horizontal and behind the vertical plane; and that the 
point projected into P’ and p” is below the horizontal and be- 
fore the vertical plane. 





If the projections P’ @’ and P’/q” intersect in R’, mie ordinate 
RM is common to both the horizontal and vertical projections. 


It is evident that if the point p’ of the vertical projection 
were given, we proceed as before to find the point pe’, and 
consequently the distance xp’ of the point from the vertical 
plane. : 


PROBLEM II. 


if the projecting planes of a given straight line be supposed to 
revolve about the projections of the straight line, till they co- 
merde with the planes of projection, vt 1s required to find the 
positions of the straight line on the horizontal and vertical 
planes. 


Let 


579 DESCRIPTIVE GEOMETRY. 


Let as be the ground line, ky 
and P’ 9’; p” Q” the horizon- : 

tal and vertical projections © 
of the lines, which are 
therefore given. From any 
two points p’, q’ of the jho- 
rizontal projection P Q 
draw P’kp”, and @'L@’, at, 
right angles to the ground 
line az, meeting the verti- 
cal projection P/Q” in v 
and qe”: also draw PR, q's 
at right angles to p’a’, and 
pr, Q’v at right angles to 
69”; and make rr, eo 8 
equal to KP", re’, and PB’ T 
Q’v equal to KP’, La’: join Rs, tv, which will be the posi- 
tions required. 

For the altitudes at p’ and q’ are, by prob. 1, equal to KP” 
and L@”, and at right angles to P’ a’ : these altitudes therefore 
must by their revolution about P’@’ coincide with the straight 
lines P’r, and q's, and consequently the straight line itself 
must coincide with rs. Exactly in the same way it is shown 
that the straight line will coincide with the line rv. 

Hence we have a simple method of determining the posi- 
tion of a straight line in space when we have its projections : 
we have only to find the pestion of the line on the horizontal 
plane by this problem, as sr} and then, , Supposing the trape- 
zoid p’rs@’ to revolve about its side p’9’ from a horizontal to 
a vertical position, the line rs will coincide with the straight 
line of which the projections are P’q’ and P”@”. 

In like manner, if we suppose the trapezoid Prva” to re- 
volve about the side e”Q” from a horizontal to a vertical po- 
sition, it will then be at right angles to the vertical plane, 
which in the construction coincides with the horizontal plane : 
and if now the vertical plane resume its vertical position, vT 
will coincide with the given line. Thus rs, tv will coincide, 
and P’@/sr, KP’Q"L will be the projecting planes of the given 
line. 

In this construction the ordinates xp”, 1@” are both altitudes 
above the horizontal plane, and therefore the perpendiculars 
PR, QS are drawn on the same side of p’a’; and because the 
points p’, @’ are both before the ground line, the perpendi- 

culars P”T, @ v_are both on the same side of vp’ @”: and in 
this case the part of the given line with which RS coincides is 
directly above the projection P’ q’. 





If 


a 


DESCRIPTIVE GEOMETRY. a3 


If the points P’ and @' of the. 
horizontal projection are both 
behind the ground line as, and 
Pe”, @’ the points of vertical 
projection, both before az; the 
construction of the positions 
RS, Tv are made as before : 
but in this case the trapezoid 
P’RsP’ must descend by re- 
Volving about p’@’, in order 
that Rs may coincide with the 
given line, which lies direct- 
ly below P’Q’. 7 

When the two points P’ 
and q’ of the horizontal 
projection are both on one 
side of the ground line, and 
the corresponding points 
P” and. @” of the vertical , 
projections are on differ- 
ent sides of acs. In this 
case we place the perpen- 
diculars P’rR and q's on dif- 
ferent sides of P’Q’; and 
ns being drawn will be the 
horizontal position of the 
given line. 7 

If we suppose the plane P’rsq’ to. revolve about P’q’ from 
a horizontal to a vertical position by the ascent of the point 
rn’, and consequently by the descent of s, ‘the straight line rs 
will then coincide with the given line. 

It is evident therefore, that the segment of the given straight 
line with which rs coincides, has one part of it above the 
horizontal plane and another part below it. 

From these varieties already considered, it appears that 
when the vertical projections P’, g” are both on the same 
side of the ground line azn, the perpendiculars P’r, q's must 
-he placed en the same side of ra’; and when Pp’, @” are 
on different sides of sn, the perpendiculars P’r, Q's must be 
placed on different sides of Pq’. 








PROBLEM II. | 
To find the poinis in which a given straight line meets the planes 
| : of projection, 
Let as be the ground line, and ra’, p’@” the horizontal and 
vertical projections of the given line, which projections are 
therefore 


Bd DESCRIPTIVE GEOMETRY. 





therefore given. Construct by prob. 2. the positions Rs, Tv, 
of the given line on the primitive planes by revolving about 
the projections PQ’, P’Q”: produce rs, tv if necessary to 
meet the projections pq, P’Q” in x and y; and x and y will 
be the points in which the given line meets the horizontal 
and vertical planes. 

If asx, rvy revolve about P’a’ and P’a”, they will (the ver- 
tical plane being supposed at right angles to the horizontal) 
each coincide with the given line, and therefore the points x 
and y are in the given line. ‘The point x belongs to the ho- 
rizontal plane, but the point y, though determined by a con- 
struction on the horizontal plane, is not a point of this plane 
through which the given straight line passes ; it is a point of 
the vertical plane, and by the revolution of the vertical plane 
from a horizontal to a vertical position, it falls below the 
ground line ap. 

In determining the points 
x and y by this method, par-. 
ticular attention must be paid 
to the positions of the points 

Re: pe Sap " : 
P,@,P, @’, with respect to a 
the ground line ag. In the 
annexed figure the points p” 
and q” fall on different sides 
of as, and therefore the per- - 
pendiculars p’r and @'s equal : 
to xP’ and 1q” must be set on different sides of P’9’, and the 
point.x in which the given line meets the horizontal plane, 
2s between the points P’ and q': but because the points P’ and 
Q 


2 










DESCRIPTIVE GEOMETRY. 578. 


a’ are both on the same side of as, the perpendiculars P’r and 


ev equal to KP’ and LQ’, must be placed on the same side of 
the projection p’@”. 


Another Construction. 


Let az be the ground line, 
and iP’, Ke”, the horizontal 
and vertical projections of 
the given line, which meet 
the ground line as in «x and 
gE. From x« and ct draw KP, 
#@” at right angles to as, 
meeting the projections LP’ 
and Ka” in Pe’, Q”: the points 
p and Q@” are those in which 
the given straight line meets 
the horizontal and vertical 
planes. 

The truth of this con- 
struction appears from prob. 
1. For the point p’ of the 


horizontal projection has p'x for its distance from the vertical 
plane, which is also the distance of the point Pp, of which 
p’ is the horizontal projection from the same plane ; but cor- 
responding to this horizontal distance xp’, there is no vertical 
distance or ordinate of p, and therefore the point P must be 
in the horizontal plane and consequently coincident with P.’ 
In like manner it appears that the vertical ordinate et has no 
corresponding horizontal ordinate, and therefore the point @, 
of which Q” is the vertical projection, must be in the vertical 


plane, and must therefore coincide with the point @’. 
If one of the projections, as 


the horizontal projection r’a’, 
be parallel to the ground line 
aB, let the other projection, 
viz. the vertical projection 
ek meet the ground line aB 
ink: from « draw xp’ at right Q’ P 

angles to aB meeting PQ’ in P; 

and P’ will be the point in which the given line meets the ho- 
rizontal plane. 

Because r’@’ is parallel to az, it is evident that a plane pass- 
ing through r’e’ and perpendicular to the horizental plane, 
and which is therefore a projecting plane of the given line, 
must be parallel to the vertical plane ; thus it appears that: 
when the horizontal projection r’a’ is parallel to the ial 

ine 








576 DESCRIPTIVE GEOMETRY. 


line as, the given line is ina plane parallel to the vertical, and 
that consequently all its horizontal ordinates are equal. 

The given line meets the horizontal plane in P’, and at any 
point @’ of the horizontal projection has an altitude equal to 
the ordinate 1@". 

If both the projections P’q’ 
and p’@” are parallel to the 2” Q” 
ground line 4B, the given line 
is in a vertical plane passing A. 
through P’a’, and in a horizon- ye , 
tal plane passing through P”9”, 0! 3 


and consequently the given dig 

line is a parallel to the ground 

line an. This also is readily deduced from finding the posi- 
tions of the given line on the horizontal or vertical plane by 
rotation about the projections by prob. @. 








PROBLEM IV. 


To find the angles which a given straight line makes with the 
planes of projections. 


rT 





Let az be the ground line, and pa’, p’@”, the horizontal 
and vertical projections of the given line. Find by prob. 2, 
the positions of the given line rs, Tv on the horizontal and 
vertical plane which produced if necessary meet the projec- 
tions Pa’, P’Q” in x and y; and axe’, Typ” are the angles 
which the given line makes with the horizontal and vertical 
planes. 

If the triangle rxP! revolve from a horizontal into a verti- 

cal 


DESCRIPTIVE GEOMETRY. © 54 


cal position it will by prob. 2, coincide with the. given line 
and the angle rxp will therefore coincide with the angle 
which the given line makes with the horizontal plane; in the 
same manner it is shown that rvp’ is equal to the angle 
which the given line makes with the vertical plane. © 

If the vertical projection P’9” be parallel. to the ground line 
ak, it is evident that rs will be parallel to ra’: of course the 
angle x will vanish, and the given straight line is parallel to 
the horizontal plane. In like manner, if P'a’ be parallel te 
aB, Tv will be parallel to P’@”, the angle y will vanish and the 
given straight line is parallel to the vertical plane. 


~ 


Another Construction. 


Let Az be the ground line, and p’z, q’k the horizontal and 
vertical projections of the given straight line which meet ap 
intandx. Draw xe’, Le’ at right angles to as, and meeting 
the projections in e’ and @" Make LM equal to Lr, and Kw 
equal to Ke”, and the angles Lmq”, KnP” are equal to the an- 
gles which the given line “makes with the hexgontal ane ver- 
tical planes. 





MOT tc Be “M ‘B 


For PY’ is-the point in which the given line meets the hori- 
zontal plane, by prob.:3, and 1@” Is the altitude of the given 
line at 1; it is evident therefore, that the angle which the: 
given line makes with the horizontal plane, is equal to the an- 
gle at the base of a right angled triangle of which the base is 
LP, OF LM. and Lash gina tse Le", and is consequently equal 
to the sy qm 


PROBLEM V. — 


To find the distsicee biebams two Lai given. byt aor Breet 
tions. 


578 DESCRIPTIVE GEOMETRY. 


Let az be the ground line, 
Pp’, @’, the horizontal projec- 
tions of the two given points, 
and Pp”, a”, their vertical pro- 
jections:. join Pr’ p", and q’ 
q”, and the straight lines pp”, 44 
q’Q", will cut the’ ground line 
at right angles in xk and L. 
Draw p’@’, and find by prob. 
2, the horizontal position rs 
of the line by means of the 
ordinates kp”, L@”. and rs will 
be the required distance. 





The truth of this construction is easily perceived ; for if 
the trapezoid p’a’sk assume a vertical position on the base 
pq’, the straight line rs will evidently coincide with the re- 
guired line. 

A similar construction may be made on the vertical plane by 
joining the points p” and Q”. 


from this construction we p* 
deduce the following theorem. 
The square of the distance 0” Y 
between two points, is equal - 
LL 


to the sum of the squares of 4. 
the interval between their or- 





dinates; of the difference of )————| x 
the horizontal ordinates, and ee 
of the difference between their S 

vertical ordinates. For if @’y, V 


@’x be parallel to an, and sv 

be parallel to a'r’, the square 

ef rs is equal the squares of R 

sv and vr; but the square of sv is equal to the square of q'P’, 
which is equal to the squares of qx, xP’, or of LK, xp’, and the 
square of va is equal to the square of yp’; therefore the 
square of Rs is equal to the squares of KL, xP’, and yP’. 


Another Construction of prob. 5. 


Let as be the ground line, and ¥’, q’, P’, @’, the horizontai 
and vertical projections of the given points as before, and con- 
sequently P’P”, qa”, at right angles to aB meeting itin kK and. 
Through q’ draw yq"z parallel to az, in which take vz equal 
to PQ’; join P’z which will be the distance required. 

To 


DESCRIPTIVE GEOMETRY. 579 


To demonstrate this 
draw x parallel to ap: 
and because yz is equal to 
P’q’, therefore the square 
of yz is equal to the 
squares of Px, qx; and 
therefore the square of 
P"z is equal to the squares 
of kL, xp’, and yp”; and 
therefore, by the preced- 
ing theorem, P”z is equal 
to the distance between 
the given points. A ig 

A similar construction may be made by drawing the parailel 


through q’ instead of @”, and using the distance P’” instead 
of p’a’. 





PROBLEM VI. 


Through a given point to draw a straight line parallel to a 
given straight line. 


Let az be the ground line, QO” 
p@’ and p’@” the horizontal = 
and vertical projections of Pp’ pot 
the given line, and r, r” the 
horizontal and vertical pro- 
jections of the given point. 
Through kr and kr” draw R's’ 
and R's” parallel to pe and 
pa’; andr's’,R’s’ will be the 
horizontal and vertical pro- 
jections of the required line. 

For let two parallel planes 
pass through P’q’, RS’, at right 
angles tothe horizontal plane; 
and let two other parallel planes pass through P’@”, p's” at 
right angles to the vertical plane which is supposed -coinci- 
dent with the horizontal plane. Conceive the vertical plane 
to revolve about az from its horizontal to its vertical position, 
while the parallel planes through P’9”, x’s” continue at right 
angles to it; then the straight line of which P’@’ and P’@" are 
the projections, will be the common section of the planes 
passing through P’9’ and P’@”; and the line of which k's’ and 
R's’ are the projections, will be in the parallel planes passing 
through these lines ; but when two parallel planes are cut by 
two other parallel planes, the common sections are parallel ; 


therefore, 





580 DESCRIPTIVE GEOMETRY. 

therefore, the straight line of which r’s’ and k's" are the pro- 
jections, is parallel to the given straight line of which the pro- 
_jections are r’Q’ and P’a” 

And because the projections k's and rs’ pass through the 
projections r’ and x”, therefore, the straight line of which r’s 
and x’ s” are the projections, passes through the given point. 

Corol. if the projections of two straight lines on the hori- 
zontal plane be parallel, and also their projections on the ver- 
tical plane, the straight line themselves are parallel. Te 
which may be added, that if two straight lines be parallel, 
their projections on the horizéntal plane are parainey and also 
their projections on the vertical plane. 


PROBLEM VII. 


‘To find the angle contained by two given straight lines meeting 
each other. 


ee Ld 


Let az be the ground line, P’e’ and PQ” the horizontal and 


vertical projections of one of the given lide, and p’r’, e’R” the 
horizontal and vertical projections of the other ; the horizon- 
tal projections intersecting in P’, and the vertical i in P’, which 
are evidently the projections of the point in which the ¢ given 
straight lines cut each other, and therefore P’xP’ 1s a straight 
line meeting the ground line as at right angles in x. 


ee R 





Find by ptob. 3, the points x and a x in which the givén 
straight lines meet the horizontal plane ; and by prob. 5, find 
‘the distances from the point of the intersection of the given 
lines to each of the points m and x; withthe centres » and N 
and distances just found, describe two arcs sx, sy intersecting 
in $5 join Ms,.Ns, and the angle msn will be the: angle re+ 
quired. 

To 


Ww 


DESCRIPTIVE GEOMETRY. 584 


To demonstrate this draw MN, and we. have two triangles on 
ihe same base mn, one of which has its vertex ins, and the 
other its vertex in the point in which the given lines intersect 
each other ; and because the two sides of one of these triangles 
is by constr ‘uction equal to the two sides of the other, and ‘the 
base mn common, therefore the angles at their vertices must 
be equal ; therefore msn is equal to the angle contained by the 
given straight lines. 

Corol. Since the triangle mp'n is the horizontal projection 
of the triangle msn, whens coincides with the point of in- 
tersection of the given lines ; it follows that the perpendicu- 
lars P’'r and st must be in the same straight line; and there- 
fore, the required angle may be found as follows: having m, 
and wn as before, join mn, and through Pp’ draw rp’s at right an- 
gles to mn, with the centre m and distance ms equal to the dis- 
tance from m to the point of intersection of the given lines, 
describe an arc sx cutting TP’s ins; join sn, and msn will be 
the angle contained by the given lines. From this construc- 
tion it is evident that if the point n go off to infinity, the 
straight line mw will become parallel to P’n, and consequently 
TP’S, “which is always at right angles to my, will then be at righi 


angles to PR’. 
Ne GS 
TS : 
saul seaiee 


RY 
\) ie 


D.¢ 

Hence the following construction of the case when one of 
the projections P’R” is parallel to the ground line as. 

Let aB be the ground line, p’@’, and Pe " the horizontal and 
vertical projections of one of the given lines, and P’R’, P’R’” 
those of the other, the vertical projection P’R” being parallel 
to as. Draw e's at right angles to P’r’ ; find m the. point in 
which the straight line of which pP’a and p’9” are the projec- 
tions meets the horizontal plane ; also find the distance from 
w to the pointim which the given lines meet each other ; with 
this distance as radius and centre m describe an arc sx meet- 
ing 








582 DESCRIPTIVE GEOMETRY. 


ing P's in S; join ms ; lastly, draw sv parallel to p'R’, arid msv 
will be equal to the angle contained by the given straight 


lines. 
KX 


| 
mt i 





When the givén lines meet each other in a point of the 
horizontal or vertical plane, the problem may he constructed 


in the following manner : 





Let as be the ground line ; Pa’, and P’a”, the projections of 
one of the given lines, and P’r’, pr” of the other ; the given 


lines meeting in the point P’ of the horizontal plane. Draw 


any 


DESCRIPTIVE GEOMETRY. 583 


any two perpendiculars to the ground line @’q” and R’r”; and 
e’, a” will be the projections of a point in one of the given 
lines, and rn’, n” those of a point in the other. Find by prob. 
5, the two distances from P’ to each of the two points of which 
@’,@", #,R” are the projections, and also the distance between 
these two points: construct a triangle of which the three 
sides are equal to these three distances, and its angle contain- 
ed by the first two distances will be equal to the angle sought. 


PROBLEM VIII. 


If a plane be given by its traces or intersections with the hori- 
zontal and vertical planes, it is required to determine tts al> 
tutude above any given point of the horizontal plane. 


Let as be the ground feu 
line, EF the horizontal | 
trace of the given plane, 
and re” its vertical trace ; 
and let P’ be any point in 
the horizontal plane: it is 
required to find the alti- 
tude of the plane above 
the point P’. 

Draw P'L parallel to r’e 
meeting the ground line in 
4; and im” at right angles aw 
to aB meeting the vertical 
trace Ec” in m’; from m” draw m’P” parallel to an, meeting px 
produced if necessary in Pp’; and P”x will be the height of the 
plane above the horizontal plane at the point P’. 

Conceive a vertical plane to pass through P’:; this plane 
will meet the given plane in a straight line parallel to er’, 
because Lr’ and EF are parallel ; and therefore, the common 
intersection of this vertical plane and the given plane is pa- 
rallel to pL, and consequently the altitude of the given plane 
above the point p’ is equal to its altitude above the point 5: 
but since the trace em’c” represents a straight line drawn on 
the vertical plane, therefore im” is the height of the point x’ 
of the given plane, that is, rm’ is the altitude of the given 
plane above the point 1 ; therefore, Lm’, and consequently its 
equal KP”, is the altitude of the given plane above the given 
point vr’. | 

Cor. It appears from the demonstration, that the altitude of 
the given plane above every point of p'L is equal to KP’ or 
um’; amd therefore, P’: and P’s’ are the horizontal and eres 

tical 





584 DESCRIPTIVE GEOMETRY. — 


tical projections of the straight line in which the vertical plane ~ 
passing through p'L intersects the given plane. — ap 


-Whenthe point Pp’ is be- 


yond the ground line as, dice 
the Strawht line pL eave . g” 
be Mes wats in a direction P 





opposite to that in the pre- 
ceding figure in order to ~ 
meet the ground line. 

For the point p' in these 
two figures, we have found 
an altitude of the given 
plane, or the plane passes 
over the given point of the 
horizontal plane. 





But in the annexed figure 
the perpendicular from 1 meets 
EG” produced below the ground 
line aB,and therefore, the given 


plane is below the horizontal hash rus 
plane at the point p,, the pes lr 5 
below P' being equal to. Kp” | 
however, the position of p’” pa 
shows universally ‘whether the ‘yi 





distance from P’ at right angles 2 
to the horizontal plane be an elevation or depression ; for 
when P” is behind the ground line, it denotes an elevation: and 
when before the gr ound line, it denotes a depression.. 


When the traces of the | 
plane are parallel to the. ~p ty 1A 9% eg, 
ground line, the construc- ¥ as r gS o 
tion is as follows: , 

Let an be the ground 
line; P’@’, PQ” the horizon- 
tal and vertical traces given 
which are parallel to as ; 
and rR’ any given puintinthe | 
horizontal plane. Through 
Rn draw Rx KT’ meeting 
RQ’, AB, P’Q" at right angles 
in x, K, and T ; make kv 
equal to xr’, join vx, and through pv’ draw r’s parallel. AB, 
meeting Vx produced if necessary ins; and xs will be the 


altitude or depression required. 





If 


DESCRIPTIVE GEOMETRY. 58 


Gr 


If the triangle vxx revolve about the side xx from a hori- 
zontal position to a vertical, it is evident that the point v will 
coincide with the point r” of the given plane, and vx will be 
the altitude at k ; therefore, r’s which also becomes perpendi- 
cular to the horizontal plane, is the altitude or depression of 
the given plane at the point x’. 

In a similar manner, when a point is given in the vertical 
plane, we may determine the horizontal distance of the point 
from the given plane. 


PROBLEM IX. 


\ 


Through a given point to draw a plane parallel to a given plane. 


Let as be the ground 
line ; EF’, eG” the hori- 
zontal and vertical traces 
of the given plane; and P’, 
pe” the projections of the 
given point. Through 
the point pe’ draw the 
straight line p’c parallel A 
to the horizontal trace 
HE, meeting the ground 
line ag inc: draw from 
c the straight line cm” at 
right angles to aB, and 
through p” the straight 
line p’m” parallel to az: 
through m” draw m’p pa- 
rallel to c’e, and pr’ parallel to er’ ; and pi’, pm’ will be the 
horizontaland vertical traces of the plane required. 

For, since the plane of which the traces are p1’, pm”, has 
em’ for its altitude at c ; therefore, by cor. to prob. 8, the alti- 
tude at p’ is also equal to cm’ or its equal xr” ; and therefore, 
the point of which the projections are p’ and P” is in the plane 
L'pm’. Suppose now the vertical plane with its parallel lines 
pm’ and gc” to revolve about az from a horizontal to a verti- | 
cal position ; and because, in this situation the two straight 
lines t'p, pm” are respectively parallel to r’n, ec’, therefore 
the plane passing through r'p, pm” is parallel to the plane pass- 
ing through r’e, nce”; and consequently, 1’p, pm” are the traces 


of the plane required, 
Vor. HT, 75 yal PROBLEM. 






586 DESCRIPTIVE GEOMETRY. 
PROBLEM X. 


To ind the straight line which is the intersection of twa given 
planes. 


\ 


Let pr’, pc” be the ho- 
rizontal and vertical traces 
of one of the given planes, 
and rF’, EG’ those of the 
other. Let the horizontal 
traces of the given planes 
meet in fF, and their verti- 
cal traces ine”. From F’ 
and «” draw F’« and e’L at 
right angles to the ground 
line aB; join F’L, Ke”, and 
FL, KG will be the hori- 
zontal and vertical traces 
of the intersection required. 

For since each of the given planes passes through the point 
F’, therefore F’ is a point in the horizontal plane through which 
the required line must pass: in like manner it is shown that 
the required line must pass through the point ¢” of the verti- 
cal plane, therefore the intersection of ; the given planes must 
pass through the points Fr’ and c” ; but F'n and xe’ are evident- 
ly the horizontal and vertical projections of a straight line 
passing through r’ and c”, and therefore F't and xe’ are the ho- ' 
rizontal and vertical projections of the straight line required. 





PROBLEM XI. 


To find the angles which a given plane makes with the fundamen- 
tal planes. 


Let as be the ground 
line, and Er’, ec” the hori- 
zontal and vertical traces 
of the given plane. In 
the horizontal trace Er 
take any point Fr’, from 
which draw r’x at right an- 
gles to it, meeting the 
ground line ap ink; draw 
Ke’ at right angles 'to AB, 
meeting £G” in e”; make 
KL equal to KF’, join o’L, 
and the angle e’Lx will be 





equal 


DESCRIPTIVE GEOMETRY. 587 


equal to the angle which the given plane makes with the hori- 
zontal plane. Or draw km at right angles to kr’ and equal to 
Ke" ; join mr’, and mr x will be equal to the angle which the 
given plane makes with the horizontal plane. | 

For, if the right-angled triangle r'xm revolve about its base 
FK from a horizontal to a vertical position, the straight line 
F/M will coincide with the given plane ;_ because xm ts now at 
right angles to the horizontal plane, and equal to ce” which is 
the altitude of the given plane above the point x. And be- 
cause the straight line er’ is at right angles to xr’, the common 
intersection of the horizontal plane, and of xr‘m in its verti- 
cal position, therefore EF’ is at right angles to mr’, as well as 
to xe’, and consequently xr’m is the inclination of the given 
plane to the horizontal plane, or xr'm is the angle required. 

Again, since G’K, KL are equal to mx, xr’, and the angles 
G’KL, MKF’ are equal, being right angles ; therefore, the angle 
GLK is equal to mr’x, and is therefore equal to the angle con- 
tained by the given plane and the horizontal plane. 

In like manner we may 
proceed to find the angle 
contained by the given plane 
and the vertical plane. 

Let az be the ground line, 
er and ec” the horizontal 
and vertical traces of the 
given plane. From any 
point-c” of ec” draw ck” at 
right angles to Ee’, and KF’ 
at right angles to an; make 
KL equal to Ke”, join F’L, and 
rLx will be the angle con- 
tained by the given plane and 
the vertical plane. 





PROBLEM XII. 


If a given plane revolves about its intersection with the horizon- 
tal plune till these two planes cownerde, rt is required to find 
on the horizontal plane, the position of any given point of the 
given plane. 


Let as be the ground line; er’ and ec’ the horizontal and 
vertical traces of the given plane ; and P’ the horizontal pro- 
jection of the given point in the plane. Draw through the given 
point p’ the straight line p’r’ at right angles to er’, and «P’L at 
right angles to P’r ; from the point x of the ground line draw 
xe’ at right angles to an; make r’L equal to Ke’; join she 

an 


686 DESCRIPTIVE GEOMETRY. 


and take in F’P’ produced 
if necessary the distances 
FM, and #'N, each equal 
to FL, and m and n will 
be the point required on 
the horizontal plane. AN 
For, if the triangle r’r'. 
revolve froma horizontal A 
to a vertical position about 
the base v’r’, tillit become 
vertical, the straight line 
F'L will coincide with the ~ 
given plane, and x with 
the given point ; because 
Ke’, to which PL is equal, 
is the altitude of the plane 
above the point pP’ of the 
horizontal. plane. And 
because, when the triangle tr’P’ is vertical, the straight line 
LF’ is at right angles to er, the intersection of the horizontal 
plane and the given plane ; therefore, if the given plane re- 
volve about er’, the given point with which x has become 
coincident, must fall on the straight line p’r’ which is at right 
angles to eF ; and since LF’ is equal to rm, or F'n, therefore 
the given point must fall on one of the points m and n. 






Another Construction. 





Let 


DESCRIPTIVE GEOMETRY. | 985 


Let as be the ground line; er’ and ee” the horizontal and 
vertical traces of the given plane ; and P’ the horizontal pro- 
jection of the given point. Having drawn as before mr’r‘n 
at right angles to er’, make FL equal to the altitude xa” of the 
plane above pv’ ; join e'x, and make F’m, Fn each equal to r’L, 
and m or n will be the point required. 


For, because the square of r’r is équal to the squares of 
pr’ and FL, that is, tothe squares of P’r’ and the altitude of the 
given point above P’ ; therefore, P’z is the distance in the given 
plane from ¥’ to the given point. And since the given point is in 
a vertical plane passing through un, therefore by the revolution 
of the given plane about er’, the point given will describe the 
circumference of acircle on the vertical plane passing through 
MN ; and this circumference must meet the horizontal plane 
an the points mM and nN; because ¥’Mm and F'n are each equal to P't. 
the distance of the given point from F’, consequently m and n 
are the points required. 


PROBLEM XIII. 
To find the angle contained by the traces of a given plane. 


Let a8 be the ground line ; me 
eF,&a@’ the horizontal and ver- G- 
tical traces of the given plane. 
Take x any given point in as, 
draw xa” at right angles to as, 
meeting the vertical trace ga” 
in G”, and e” will be a given 
point in the given plane, and k 
ts horizontal projection : find 
by prob. 12, the position 1 of 
the point c” by revolution of 
the given plane till it coincide 
with the horizontal | plane ; 
join ex, and P’ EL will be the 
angle sought. 





For, since the point c” falls on the point 1 of the horizontal 
plane, therefore the straight line or trace ra” will coincide 
with £L; but Fx is the Haron trace, therefore FEL is 
the angle contained between the traces of the given plane. 


\ 
When 


590 _ DESCRIPTITE GEOMETRY. 


When one of the angles arc” 
is acute and the other arp’ 
obtuse, take as before’ any 
point x in the ground line 
as, make xe” perpendicular 
to AB, meeting the vertical 
trace EG” in e’; and conse- 
quently, c” is a point of the 
vertical plane, of which kis AM = K 
the horizontal projection: ‘find 
tthe position of c” on the 
horizontal plane ; Join BL; 
and Lep’ will be ‘the angle 
contained between the traces 
of the given plane. 

If the ‘given horizontal 
trace Ep, instead of being 
given before the ground line as, is given behind it as EF’, the 
construction is exactly the same as before ; and LEF’ is the an- 
gle contained between the horizontal trace ur’ and the ver- 
tical trace Ea’’, 






\ 
} 


PROBLEM XIV.| 
\ 
To describe a plane through three given points. 


Let as be the ground 
line; Pp’, @’, n the hori- 
zontal projections of the 
three given points ; and 
ep”, a’, rR”, their corres- 
ponding vertical projec- 
tions- Through P , @ ; 
and pv”, e” draw arr, 
and p’@’a”, the horizon- 
tal and vortieal projec- 
tions of a straight line 
passing through two of 
the given points; and 
through qn; Q's é! draw 
Qn H’, and QR” the hori- 
zontal and vertical pro- 
jections of a straight 
line passing through o- 
ther two of the given points. Find by prob. 3, the points r’ 
and #’ in which these two straight lines meet the horizontal 

plane, 





DESCRIPTIVE GEOMETRY. 591 


plane, and a” the point in which one of them meets the ver- 
tical plane. | : 

Draw uF’e meeting the ground line aB in £; join Ec”; and 
EH, EG’ will be the horizontal and vertical traces of the re- 
quired plane. 

The straight line of which r’@’ and Pa” are the horizontal 
and vertical projections, passes through two of the given points; 
therefore, that straight line must lie in the required plane ; 
but that line passes through the point Fr’ of the horizontal 
plane; therefore, the required plane passes through F’: for the 
same reason the required plane must pass through u’; and 
consequently, er’H is the horizontal trace of the requir- 
ed plane: and because the vertical trace must pass through 
the points & and c” of the vertical plane, therefore rn” is the 
vertical trace ; and consequently, EF and ec” are the horizon- 
tal and vertical traces of the required plane. 

Instead of finding two points in the horizontal and one in 
the vertical plane, we may find two in the vertical and one 
in the horizontal plane: or we may find two points in the 
horizontal plane, and two in the vertical ; and the two straight 
lines drawn, these points on the horizontal and vertical planes 
will be the traces of the required plane. Sometimes the 
point = in which the traces meet the ground line is too re- 
mote to be conveniently found in practice: this happens 
when one or both of the traces are nearly parallel to the 
ground line: in this case, it is proper to find two points 
through which the horizontal trace must pass, and in a simi- 
lar manner, two points through which the vertical trace must 
pass in the vertical plane. 


PROBLEM XY. 


Through a given point to draw a straight line perpendicular 
to a given plane. 


Let a8 be the ground line; Er’, rc” the horizontal and ver- 
tical traces of the given plane; and P’, e” the horizontal and 
vertical projections of the given point. Through the projec- 
tions P’ and Pp”, draw the straight lines P’ F’, P’ c” at right an- 
gles to the given traces cr’ and Ec’, and rr’, p’a” will be the 
horizontal and vertical projections of the line required. 

Conceive a vertical plane to meet the horizontal plane in p’r’; 
and because er’ is drawn in one of these planes at right angles 
to their common section, therefore rr’ is perpendicular to 
the vertical plane passing through rr’; and consequently, the 
given plane which passes through er’, is at right angles to the 

| vertical 


“ey 


“< 


592 DESCRIPTIVE GEOMETRY. 


vertical plane passing through r’r’: but the given point is in 
this vertical plane, therefore a straight line drawn from the 
given point at right angles to the given plane, must lie in this 
vertical plane, and therefore, r’r’ is the horizontal projection 
of the required straight line. In a similar manner, it may be 


on 


shown that pc” is the vertical projection of the required line. 





Corol. From the demonstration of this construction, ii ap- 
pears that when a straight line and a plane are at right angles 
to each other, the projections of the straight line are at right 
angles to the corresponding traces of the plane. 


» PROBLEM XVI. 


Through a given point, to draw a plane at right angles to a 
given straight line. 


Let az, be the ground line; mr’, ne” the projections of the 
given straight line ; and r’, e” the projections of the given 
“point. 

Through the point P’ draw ur's at right angles to MF’ meet- 
ing the ground line in 1 ; draw from x the straight line Lp” at 
right angles to AB, meeting the straight line P” p” parallel to 
AB in D’: Through p’ draw ep’e” at right angles to the ver- | 
tical projection no”, meeting aB in E; from E draw EF at right | 
angles to the horizontal projection MF, and EF, EG” are shove 
traces of plane required. 

Because the traces er’, ec” are at right angles to sahe given 
projections mr’, na” of the straight line ; therefore, by the pre- 
ceding corollary, the plane of which poe EG’ are the traces 

is 


DESCRIPTIVE GEOMETRY. 593 





1s at right angles to the given straight line. Again, because 
KP” is equal to tp”, which is the height of the plane above the 
horizontal plane at L, and consequently to the height at P or 
any point in LPH ; therefore, the plane of which er’, gc” are 
the traces, passes through the given point, and is consequently 
the plane required. 


PROBLEM XVII. 


Tt is required to find the pont in which a given straight line in- 
tersects a given plane. 


Let ae be the ground line ; r’@’ the horizontal, and p’1 the 
vertical projection of the given line; and Er’, ce” the horizon- 
tal and vertical traces of the given plane. 

Let v’@’ meet the ground line az in m, draw me” at right an- 
gles to an, and having drawn mr at right angles to r’e’, make 
it equal to me”; let Pe’ meet Ee’ in F, and join F’R. Again, 
find by prob. 2, the position @’r of the given line on the ho- 
rizontal plane by revolution about the horizontal trace mr’, 
and let QT intersect F’R ins; draw sp’ at right angles to @’F, 
and r’xp” at right angles to aB, then P’,e” will be the horizon- 
tal and vertical projections of the point in which the given 
plane is intersected by the given straight line. 

Because mn is equal to me”, it is evident, that F’R is the ho. 
rizontal position of the straight line in which the given plane 
is cut by a vertical plane, passing through mr’, and @’r is the 
horizontal position of the given line; therefore, if the trian- 
gle qs¥F’ revolve about qr from a horizontal to a vertical po- 
sition, the point s will be at once in the given straight line and 
given plane ; and will therefore be in the point required. 

Vor. If, 76 And 


594 DESCRIPTIVE GEOMETRY. 





And since, sp’ is perpendicular to P’a’, the straight line r’s 
will become perpendicular to the horizontal plane ; and there- 
fore, P’ is the horizontal projection of the required point. 

Lastly, because re” is the vertical projection of a pomt in 
the given line of which P’ is the horizontal projection ; there- 
fore, r’ and e” are the projections of the required point on the 
horizontal and vertical planes. 


7 


Another Construction. 


Let az be the ground 
line; Pq’, Pa” the horizon- 
tal and vertical projections 
of the given line ; and Er, 
E@’ the horizontal and ver- 
tical traces of the given 
plane. 

Let r’e’ meet aB in M, and 
EF in fF; and from m and 
F’ draw MN, FL at right an- 
gles to the ground line as, 
meeting ee”, and aB inN 
and L ; join Ln” meeting Q” 
o” produced if necessary in P’. Draw P’Kr’ at right angles to 
AB meeting @’F’ in P’, and Pr’, p” will be the horizontal and ver- 
tical projections of the point required. 

- For mn" is the altitude of the given plane above the point m, 
and nw" is the vertical projection of the point of the plane in 
which it is intersected, by the perpendicular to the horizon- 
tal plane at m: also 1 is the vertical projection of the point 

F’, 





DESCRIPTIVE GEOMETRY. 596. 


yr’, and therefore, in” is the vertical projection of the straight 
line which is the common section of the given plane, anda 
vertical plane passing through mr’: but the given line is in 
this vertical plane; and therefore, meets the given plane in 
some point ef the line, in which the given plane ts intersected 
by the vertical plane passing through mr’: therefore, the ver- 
tical projection of the required point must be in un”, and it 


wes 


must also be in Q’c”; therefore it is in Pp’, the intersection of 
Ln”, Q’e"; but P’ is the corresponding horizontal projection 
of that point in the given line of which Q” is the vertical pro- 
jection, and consequently r’, e” are the horizontal and vertical 


projections of the point required. 


PROBLEM XVIII. 


To draw through a given point a straight line perpendicular to 
@ given straight line. 


Let an be the ground line ; 
DF, cH” the projections of . . v 
the given straight line, and Ae ; 

P’, P’ the projections of the 
given point. Find by prob. 
16, er’ and en’ the horizon- 
tal and vertical traces of a 
plane at right angles to the 
given straight line, and 
passing through the given 
point of which the projec- 


tions are P’ and p’; again _& va 
by prob. 17, find @’ and Q”, on 
the projections of point in ‘ 


which the given line meets 

the plane of which er and 

EH’ are the traces: lastly through Pr’, 9’, P”, 9” draw P’@’, 
P’@”, which will be the horizontal and vertical projections of 
the line required. 

For, because the plane of which er’, ex” are the traces is 
at right angles to the given line, and passes through the given 
point of which P’, pe” are the projections ; therefore, the per- 
pendicular on the given line from the given point must be in 
the plane F’ex”, and must pass through the point in which the 
given line passes through this plane, that is, through the point 
of which @ and Q" are the horizontal and vertical projections : 
the line required must also pass through the given point of 
which P’, @” are the projections ; consequently, Pe’, PQ” are 
the horizontal and vertical traces of the line required. 

” PROBLEM 





596 DESCRIPTIVE GEOMETRY. 
PROBLEM XIX. 


lo find the angle contained between a given plane and a giver 
straight line. 


The construction may be given in general terms as follows : 
Take any point in the given straight line, from which by prob: 
15, draw a straight line perpendicular to the given plane. 
Find by prob. 7, the angle contained by these two straight 
lines, which will be the complement of the angle required. 

For, if we draw.a straight fine inthe given plane, from 
the point in which the given line meets it to the point in which 
the forementioned perpendicular meets it, we shall have a 
right angled triangle in which one of the acute angles is the 
angle contained by the given plane, and given straight line ; 
and the other acute angle, is the angle contained by the given 
straight line and the perpendicular onthe plane: and there- 
fore, this latter angle is evidently the complement of the an- 
gle required. 

I shall now illustrate this general construction by a parti- _ 
cular example in which all the operations are exhibited. 

Let as be the ground line, er’, ee” the horizontal and 
vertical traces of the given plane; and P’q’, P’Q” the horizon- 
tal and vertical projections of the given straight line. From 
Q" the point in which the vertical projection p’e” meets the 
ground line aB; draw Q"@’ at right angles to a8, and q@’ is the 
- point in which the given line meets the horizontal plane. 
From @’. 9” draw @’'F’, ee’ at right angles to EF’, ne”, and @'F’, 
g"a” will by prob. 15, be the projections of a straight line 
drawn from the point of which @’, e" are the projections at 
right angles to the given plane. 





Novw 


DESCRIPTIVE GEOMETRY. 597 


Now, make r’c equal to tr’ which is the height of the per- 
pendicular to the plane at Fr’, and @c is the distance of the 
poiut q’ from the point of which the projections are F’ and F” ; 
again, make p’n equal to KP” which is the height of the given 
line at pv’, and consequently Q'n is the distance of the point @ 
from the point of the given line of which Pp’ and P” are the 
projections. Through p’ draw pp’H parailel to an, make pH 
equal to p’r” and ra will be the distance between the two 
points of which the horizontal projections are P’, F’, and their 
vertical projections P” andr”. Lastly, with the centres F’ and 
H, and distances r’m and Hm equal to e’c, and @’n, describe two 
arcs intersecting in m, and F’mu 1s the angle contained by the 
given straight line, and the perpendicular to the plane having 
QF’ and q’a” for its horizontal and vertical projections, and 
is therefore, the complement of the angle required. 


PROBLEM XxX. 
To find the angle contained by two given planes. 
General Construction. 


Find by prob. 10, the straight line which is the common 
section of the given planes, through any point of which draw 
by prob. 16, a plane at right angles to it: by prob. 10, find 
the common sections of this plane with each of the given 
planes; and lastly, by prob. 7, find the angle contained by 
these two intersections, which will be the angle required. 

Because a plane cuts the common section of the given 
planes at right angles, it is evident, that the intersections of 
this plane with the given planes are at right angles to the 
common section of the given planes ; and consequently, the 
angle contained by these intersections is equal to the required 
angle contained by the given planes. 

In the figure annexed, all the lines are drawn that are 
wanted in the construction ; let an be the ground line; pr’, 
pe’ the horizontal and vertical traces of one of the given 
planes ; and er’, ec” those of the other. 

Draw Fm, ¢’c at right angles to az. and F’a, ma” are the ho- 
«izontal and vertical projections of the common section of the 
given planes: draw r’L,in’ at rightangles to cr’, o’M, andr’ Liy” 
are the traces of a plane at right angles to the common section 
of the given planes. Draw u’n at right angles to as, and F'H, 
mH are the projections of the intersection of the plane y’LA’ 
with the given plane r’pu”: and uc being made equal to ar’, 
wc is the distance in this intersection from rF’ to 8”; in like 
manner by making KN equal to kr’, we have xn equal to the 

distance 


598 DESCRIPTIVE GEOMETRY. 





distance from F’ to x” in the intersection of the plane r'Lw’; 
and the given plane rec", and ux” is the distance between 
the points a” and «” which are both in the vertical plane ; 
now with the centres Hn”, kK’ and distances n’c, xn describe 
two arcs intersecting in x, and k"xu’ will be the angle required. 


PROBLEM 21. 


Through a given straight line to draw a plane parallel to an- 
other given straight line. . 


Through any point of the line through which the plane 
must pass, draw by prob. 6, a straight line parallel to the 
other given line. Find by prob. 3, the points of the horizon- 
tal plane in which this parallel and the first mentioned line 
pass through it; through these two points draw a straight 
line, which will be the horizontal trace of the plane required. 
_ In like manner, by finding the two points.in which the paral- 
lel and first mentioned line meet the vertical plane, and draw- 
ing a straight line through them, we have the vertical trace 
of the plane required. - 

Because the plane thus constructed passes through the 
points in which the first mentioned line and parallel meet the 
fundamental planes ; therefore, these lines are in the plane ; 
but when two straight lines are parallel, any plane passing 
through one of them is parallel to the other. 


Construction in which are drawn all the necessary lines. 


Let ap be the ground line; rr, P’c’ the horizontal and 
vertical projections of the straight fine through which the 
plane 





Note. The line cx” at right angles to aB is wanting in the figure. 


DESCRIPTIVE GEOMETRY. 599 


plane must pass, and xx’, nx” the horizontal and vertical traces 
ef the straight line to which the plane must be parallel. 





Draw any straight line r’p” at right angles to aB, meeting the 
horizontal and vertical projections P’r’ and P’e” in P’ and p”; and 
p’, P’ will be the projections of a point in the first mentioned 
line. Through rp’ and Pe” draw ne'n”, mp’H” parallel to the 
projections kx, ux” of the second given line, and nn”, mu”, are 
the projections of the straight line passing through the point 
pr’, P’ and parallel to the straight line xx’, Lu’. Find r’ and 
n’ the points in which the line r’r’, p’e”, and the line nw’, wx” 
meet the horizontal plane, and 6”, n” the points in which the 
same lines meet the vertical plane ; and eEr'n’ cen” being 
drawn will be the horizontal and vertical traces of the plane 
required. | 


CHAPTER 


[ 600 j 


| CHAPTER III. 
CoNSTRUGTION OF THE Cases IN SPHERICS: 
Right-angled Spherical triangles. 

CASE Tf. 


Given the two legs of a right angled spherical triangle to sind the 
: the angles and hypothenuse. 


Let as be the ground line, 
AEF’, AEG’ two angles of which 
the measures are the given 
sides. Find by prob. 11, the 
angles which the plane having 
BF and ea” for its horizontal 
and vertical traces makes with 
the horizontal and vertical — 
planes, and these angles will be 
the angles required. 

Thus xxe” is the angle of 
the spherical triangle which is 
epposite to the side that mea- 
sures the angle arc’. Again 
find by prob. 12, the position 
m on the horizontal plane of 
the point «c” in the vertical 
trace ec”, and rem will be the 
angle of which the hypothe- 
nuse is the measure.) Bh 

For, conceiving xec to be at right angles to the horizontal 
plane arr’ in the common section as, and that a plane passes 
through er’ and ze”, we shall evidently have a solid angle at rE, 
of which the three plane sides are ker’, kEG’, and the angle 
of which the sides are er’, re” and which by construction is 
equal to the angle r’em. This solid angle at & has its sides 
and the inclinations of these sides equal to the sides and angles 
of a right-angled spherical triangle; the right angle contamed 
by the horizontal and vertical planes, being the right angle of 
the spherical triangle, and the inclinations of the plane passing 
through EF’ and Ec”, to the horizontal and vertical planes being 
the oblique angles of the triangle. 





CASE 


DESCRIPTIVE GEOMETRY. €01 


CASE II. 
Given one leg, and the hypothenuse of a right-angled spherical 
triangle to find the three remaining ead 


Let aB be the ground 
line ; axe’, rem the given 
leg ‘and "hypothenuse. 
Draw from any point x of 
the ground line ar the 
straight line Kr’M at right 
angles to er: make ki 
equal to xr’; draw ko 
at right angles to az, and 
make ie equal to r’m; then 
shall er’, ne” be the hori- 
zontal and vertical traces 
of the plane of the hy- 
pothenuse, arc” the leg 
required, and c’ix the 
angle adjacent to the giv- 


en side or leg arr’. 
This construction is merely the converse of that i in case 1 


Beéause ie” is by construction equal to F’m, therefore m is the 
horizontal position of the point e’ by the rotation of the plane 
of the hypothenuse about er’; and therefore, the angle con- 
tained by F's and ec”, when considered in the vertical plane, is 
equal to the given hypothenuse Fem, Also, by prob. 11, 6’LK 
is the inclination of the plane of the hypothenuse to the hori- 
zontal plane. By the same prob. we may determine the in- 
clination of this plane of the hypothenuse to the vertical 
plane, which will be the angle of the spherical triangle ae 
cent to the side are”. 





CASE III. 


Given one leg and the adjacent angle of a right-angled irang te 
to determine the remoning parts. 

Let az be the ground line; and arr’ the given leg on the 
horizontal plane. From any point K in thé ground line a# 
draw xr’ at right angles to er’; make Kx equal to xr’ 3 at L 
make the angle ate’ “equal to the given angle, and let Le” ‘meet 
the straight line Ke” at right angles to aB in G”; join ee”, and 
EF’, EG’ will be the horizontal and vertical traces of the plane 
of the hypothenuse ; also, if rm be made. equal to ie” the 
angle Fem will be the hypothenuse. 

For, since xe" is at right angles to AB, KF to EF’, and KL equal 
_to xr’; therefore the given angle xia” is by prob. 11, the in- 

clination to the horizontal plane of the plane of which EF” 
and ec” are the horizontal and vertical traces, which is the 

Vou. U, 77 spherical 


602 DESCRIPTIVE GEOMETRY. 
Gr : 





spherical angle contained by the hypothenuse and base. 
Therefore xec” is the remaining leg: and, since m is the hori- 
zontal position of the point ¢” by rotation about EF’, therefore 
FEM is the hypothenuse. 
CASE IV. 
Given one leg and the opposite angle to find the remaining ie 
of the triangle. 


Let as be the ground 
line ; and arc” the an- G” 
gle of which the given | 
leg is the meusure. 
From x, any point in 
AB draw xe’ at right 
angles to aB, and make 
KG’L equal to the com- 
plement of the given 
angle : With the cen- 
tre k, and distance KL 
describe a circular arc 
LFH and draw EF to 
touch the arc Lr’H in 

, then er’, Ec” are 
the horizontal andver- 
tical traces of the plane 
of the hypothenuse, 
and uf course the angle 
4eF has for its mea- 
sure the remajning leg of the triangle. 





DESCRIPTIVE GEOMETRY. 603 


For, if we join kr’, the angle xr’r is aright angle, and there- 
fore xie” is the angle made with the horizontal plane by the 
plane of which the horizontal and vertical traces are er’, Ec’. 

The hypothenuse may be determined as before. 


CASE V. 


Given the hypothenuse and one angle of a spherical triangle, to 
determine the remaining parts. 


Let r’em be the giv- 

en hypothenuse and G* 
from any point F in the 
side eF draw ¥F'M at 
right angles to F'g: at F’ 
make mr’H equal to the 
given angle, and draw 
mH at right angles tor’H, 
In mr’ produced take 
F’K equal to F’'H; join 
EK, and draw xe” at 
right angles to EK; 
make Kx equal to xr’, 
and Le” equal to rm; 
join Ee”, and KeF’,xEc” 
will be the angles of 
which the required 
legs are the measures. 
For if as be the 
ground line ; and er’, 
eq’ the horizontal and 
vertical traces of a plane, it is plain that the inclination.of this 
plane to the horizontal plane is xxce”, which by construction 
is equal to FH: and the hypothenuse or angle contained by 
e¥’ horizontal and Ea" in the vertical plane is equal to the 


given angle Fem. 





M 


CASE VI. 


Given the angles of a right angled spherieal triangle to deter- 
mine the legs and the hypothenuse. 


Let az be the ground line ; with one of the given angles xe" 
as an angle and the complement KEG’ ot the other given angle 
as an opposite side, construct a spherical triangle by case 4, 
and the: complements of its remaining side ser’, of its hypo- 

-thenuse F’em, and of its remaining angle, will be the hypothe- 
nuse and two legs of the triangle required. bs 
et 


804 DESCRIPTIVE GEOMETRY, 


G" 





Let asc be an equilateral, 
equiangular, rectangular spheri- 
cal triangle, all the sides being 
quadrants and all the angles right 
angles; and let any two great 
circles BFD AFE pass through B 
and a, and Bp, ag will be quad- 
rants, and the angles at p and £ 
right angles ; and therefore apr, 
Ber are right angled triangles, 
having the angles at r equal ; and omer 
the remaining Heals of. each are D 
complements of those of the other: | 
thus, BF is the complement of pr; BE is the complement of 
ck or of the angle par, and the angle ExF or its measure pe 15 
the complement of the side ap. 

Suppose now that the angles par, pra are given to find the 
remaining parts of the triangle apr, From what has just been 
shown, we have in the triangle ser, the angle Bre and the 
opposite side sz which is the complement of the given angle 
par. It is evident therefore, that when two angles are giyen 
in the triangle sanz, we have only to construct the triangle 
ger of which the angle ere and the opposite side Be are given 5 
which is effected as in case 4. And afterwards we obtain 4F, 
ap, DF by taking the complements of re, the angle esr, and 
the hypothenuse sr. : | “hi 





CASES 


DESCRIPTIVE GEOMETRY, 605 


CASES OF OBLIQUE ANGLED SP EMICAL 
TRIANGLES, 


— 
CASE I. 


Given the three sides of a spherical triangle to determine the 
angles. 


With the centre & and any radius ag, describe on the hori- 
zontal plane a circle ascp, in the circumference of which 
take aB, Bc, cp equal to the three given sides: from a and 
p draw aFH and pKL at right angles to ep and ec. intersecting 
each other ine. On the chord au as a diameter describe the 
semicircle amu ; drasy em at right angles to aH; join FM, and 
MFH will be the angle contained by the sides equal to az and Bc. 


D 





Supposing x to be the centre, and at the radius of the 
sphere, te which the spherical triangle belongs ; let the se- 
micircle amy revolve from its horizontal to a vertical position, 
and suppose the angle BEA to revolve about the side Br; it is 
_ plain, that the point - a will describe the circumference amy. 
In the same mannér, the point p, will describe the circumfer- 

ence 


606 DESCRIPTIVE GEOMETRY. 


ence of a vertical circle on the diameter pr. It is evident 
therefore, that mc will thus become the common section of 
these two vertical semicircles, and m the point with which a 
and p coincide, when Ea and Ep are coincident. And because 
when the triangle Gmr is vertical, ur will be at right angles to 
EB, the angle crm will be the inclinotion of the plane of the 
angle pea to the horizontal plane when the points a and m co- 
incide. | 

By a similar construction, we may determine the remaining 
angles. 


Another Construction. 


Let az be the ground line; in which take any point c, and 
and draw cp” at right angles to as. At any point pv” in this 


woe 


perpendicular make the angles ep’g, cp’F, and rp’e” equal to 


Dp” 


: G’ 





A 


the angles of which the given sides of the spherical triangle 
are the measures: make p’c” equal to p’E; join Fe’; and with 
the centres c and F, and distances cr, rc” describe two arcs 
intersecting in H’; join cu’, and rcu’ will be the angle opposite 
to the side which measures the angle rp'c’. 

Conceive the vertical plane to stand at right angles to the 
horizontal, and in this situation suppose px” te be joined ; also 
draw rH. Nowcop’r being at right angles to the plane cru’, 
it is evident that cru’p” is a triangular pyramid of which the 
base is cru’, and vertex p” : and since cu’ is equal to ce, and 
the angles p’cx’, pce equal being right angles, therefore the 
angle cp’H’ is equal to cp’r, and p’H’ to p’e. Also, because 
pH’ is equal to p’r, that is, to pe”, and ru’ to Fc”, and p’F comn- 
mon to the two triangles rp’n’, Frp’c’; therefore the angle 


Fp'H is equal to the given angle rp’c’; and thus it is manifest, 
that 


DESCRIPTIVE GEOMETRY. 607 


that the three angles at the summit pv” of the pyramid, viz. 
ep’. cb’ F, Fp’H’ are equal to the three given angles cn’r’, 
cpD’F, FD’«’; consequently, if with the centre p’ and radius p’r 
we describe a spherical surface, its intersections with the 
planes of the angles cp’H’, cp’F, rp’H, will be the spherical 
triangle, cLK having the given sides. And because p’c is at 
right angles to cr aad cn’, therefore cr and cw’ are the tan- 
gents of the arcs cx and ck, and therefore the plane angle rcx’ 
ig equal to the spherical angle icx. 


CASE If. 


Given two sides and the contained angle to determine the remain- 
ing purts of the triangle. 


With the centre © and any distance ra, describe on the 
horizontal plane, the circle ancy; in the circumference of 
which take aB, Bc equal to the two given sides. Make sx 
equal te aB ; join aH, and on it as a diameter describe the se~ 





micircle ama: at the centre r make the angle urm equal to 

the given contained angle ; draw me at right angles to au, and 

_ akp at right angles to cx, and cp shall be the side of the sphe- _ 
rical triangle that is opposite to the given angle. 

The 


608 DESCRIPTIVE GEOMETRY. 


The truth of this constriction is evident from the demon- 
stration of the first construction given to the preceding case. 


Another Construction of Case 2. 


_ Let an be the ground line ; in which take any point c, and 
draw cp” of any length at right angles to As. Make the angles 
cb’E, cD'F equal to the two given sides, and ecu’ equal to the 
given angle: with the centre ¢ and distance cr describe the 
arc EH’, and join rH’; lastly. make p’c", re” equal to pe and 


rH respectively, and the measure of the angle rp’e’ will be 
the side required. 


15 a Ves Ag 





This construction is evident from the demonstration of the 
second construction given to the preceding case. 


CASE IIL. 


Given two sides and an angle opposite to one of them to find the 
remaining side. . 


With the centre e’ and any radius c's’ describe on the hori- 
zontal plane a circle agp’, in the circumference of which take 
cB, BD’ equal to the given sides, and c’a equal to Bc’. Join as, 
on which as a diameter, describe the semicircle ac’B: at F 
the centre of 4c’B make the angle arc” equal to the given 
angle ; from ce” draw e’H at right angles to aB, and make ck 
equal to the chord sp’ ; with the centre # and radius xu de- 
scribe the circle kim, cutting ace in Lv and m’,and cL, or cm 
will bé the required side of the triangle. mr 

Conceive az, to be ihe ground line ; and ac”s to be on the 
vertical plane, making right angles with the horizontal plane ; 
also, suppose a sphere to be described with the centre x and 

distance 


DESCRIPTIVE GEOMETRY, 609 


distance Ec’; then it is evident that the semicircumference 
ac’s, is in the surface of the sphere, that c’ is one of its poles ; 
and, consequently, that the distance from c’ to «” in the are of 
a great GCE of the sphere, is equal to the arc c's. 





Imagine a plane to pass through c's’, and make with ar an 
angle equal to the given angle ; this plane will evidently pass 
shrough the point «” of the vertical plane, because the angle 
«©’FH was made equal to the given angle, and the arc of the 
great circle between c’ and 4” Jying in the forementioned plane, 
will make with c’'a an angle equal to the given angle. 

Again, e’# being at right angles to the horizontal plane, 
every point of the circumference Kim’ is equally distant from’ 
a’; consequently the arc of the great circle passing from ¢” to 
x or M has its chord equal to c’n, or the chord sp’, and there- 
fore the arc between ¢” and 1, or between c’ and m, is equal 
to the given side Bp’. 

Thus it is plain that ce” and’ or m’ are the angular points 
of a spherical triangle having two of its sides equal to c's and 
sp’, and its angle opposite the side c’r’ or c’m’ equal to the given 
angle ; ; and therefore, c'L’ or c’m’ is the third side required.. 

The three remaining cases of oblique angled spherical tri- 

-angles are constructed exactly as the three preceding cases 
by means of the polar triangle described, vol. ii, page 30. 
See the author’s figure to theorem iv of spherics. 

Vou. Il. 78 CASE 


610. DESCRIPTIVE GEOMETRY. 


CASE IV. 


If the three angles of the spherical triangle per be giver 
to find the sides ; we take the supplements of the given angles, 
we have the three sides of the supplemental or polar triangle 
asc ; and, the angles of the triangle being found by case 1, their 
supplements will be the sides required in the triangle per. 


CASE V. ~ 


If the side pz and the adjacent angles at p and x, be given 
to find the remaining parts of the triangle per ; by taking the 
supplements of the given parts we have the two sides as, ac 
and the contained angle ac, to find the remaining parts of the 
triangle azc by case 2. 


CASE VI. 


If the side pr and the two angles at p and £ be given in the 
triangle per to find the remaining parts ; we have in the trian- 
gle anc the two sides aB, ac, and the angle asc opposite to on 
of them, to find the remaining side by case 3. 


OHAPTER 


oe ss Bs ee 
CHAPTER IV. 


CONSTRUCTION OF THE CONIC. SECTIONS, 


PROBLEM I. 


To construct a conic surface. 


1. Draw the ground line as, in the horizontal plane; take 
any point p’ for the centre of the circular base of the cone, 
and with the radius of the base describe about P’ as a centre, 
the circle cq’r’ for the base of the cone. From P’ let fall on 
ap, the perpendicular r’x, in which produced take xP” equal 
to the axis of the cone; that is to the distance between the 
vertex of the cone, and the centre r’ of the base ; and P” will 
be the vertical projection of the vertex of the cone. 





Because the axis of the cone is at right angles to the base, 
it is evident that the horizontal projection of the axis is sim- 
ply the point P’ ; and xr” at right angles to the ground line az 
is its vertical projection: and therefore, P’ and P” are the ho- 


rizontal and vertical projections of the vertex of the cone. 
, 2, Ta 


612 DESCRIPTIVE GEOMETRY. 


2. To project the slant side of the cone, take any point «’ 
in the circumference of the base, through which and the cen- 
tre p’ draw @’p’r’ a diameter of the base ; also, from @’ draw 
qt at right angles to the ground line aB; and join e’L ; then 
will e’@’ and P’t be the horizontal projections of the slant side 
of the cone which passes through the point @’. 

This construction of the slant side is evident, because 1 
being the vertical projection of the horizontal point q’ ; there- 
fore pr’ and @ are the horizontal projections of two points of 
the slant side, and P’ at the corresponding vertical projections ; 
and consequently, PQ’, Pt are the required projections of the 
slant side passing through @’. 

If we make a similar construction for the slant side passing 
through r’ we have the construction of the two slant sides of 
the cone, in which the curve surface of the cone is inter- 
sected by a vertical plane passing through the axis of the cone. 

Thus pe’ and e’m are the vertical projections of the opposite 
slant sides passing through the extremities of the diameter 
a'r’ of the base, and the opposite radii P’@’, P'R are the cor- 
responding horizontal projections. 

3. To find the vertical. projection of any point of the sur- 
face corresponding to any given horizontal projection. Let 
s’ be any given horizontal projection of a point of the curve 
surface of the cone; draw the radius r’s’q’, end having con- 
structed the slant side by its projections P’@’, p’L, draw s'ns” at 
right angles to the ground line as, meeting the vertical pro- 
jection P”L in s", and s” will be the vertical projection of that 
point of the conic surface which has s’ for its horizontal pro- 
yection. 

In the preceding construction we have considered only that 
part of the whole conic surface which is between the vertex 
and base ; but as the conic surface may be extended indefi- 
nitely downwards below the base, and upwards above the ver- 
tex, it is plain that the horizontal projection R'P’q’ of the op- 
posite slant sides as well as the vertical projections P’K, P’™, 
should be produced indefinitely both ways ; that 1 is, RQ to- 
wards F’ and J’; and P’x, P’m towards BE’, R’; H’,D’. 

Now, the vertical projections r’e”, p’u” being both in a ver- 
tical Plane passing through FJ’, if we produce s's’ to meet 
p’H” in Tt", we shall have rv” for the vertical projection of the 
point in which a perpendicular to the horizontal plane 
ats’ meets the slant side which passes through the point r 
of the base: this perpendicular therefore meets the conic 
surface in two points, of which s° and 1” are the vertical pro- 
jections, the horizontal projections being coincident in the 
points’. Also, the socbeiay ordinates of these two points being 

NS’ 


DESCRIPTIVE GEOMETRY. 613 


ns” and wr’, itis evident that the part of the perpendicular at 
s' which is projected into s’r", falls without the conic surface ; 
the remaining parts of it falling within the upper and lower 


divisions of the conic surface. ‘ 





If we produce qx to meet px” in Q”, the point @” will be 
the vertical projection of the point in which the perpendicu- 
Jar trom @ to the herizontal plane, meets the upper division 
of the coni¢ surface. In like manner, if rx be produced t¢ 


614, DESCRIPTIVE GEOMETRY. 


Rr’, we have the vertical projection of the point in which the 
per pendicular at k’ meets the slant side of the cone passing 
through @’: and, because mr” and K@” are equal, as is evident 
from the construction, it follows that-n” and @” are the vertical 
projections of two points diametrically opposite in a circular 
section of the upper conic surface parallel to the base. 

If we take any point F’ in R’@’ produced, and draw p’F’, oF" 
at right angles to the ground line as, it is plain that p” and g£” 
are the vertical projections of the points in the upper.and 
lower divisions of the conic surface through which a straight 
line passes, that is, perpendicular to the horizontal plane at F’, 
so that or’, op’ are the horizontal and vertical ordinates of the 
point of intersection in the upper division, and DF’, or” in the 
lower. Also that part of the perpendicular at r’, that is re- 
presented by p’r” , falls without the conic surface ; ‘and the re- 
maining parts above p” and below x’ ’ fall within the upper and 
lower divisions of the conic surface. 


PROBLEM II. 


To find the point in which a given plane ts cut YY the given slant 
side of a cone. 


Pp” 





DESCRIPTIVE GEOMETRY. 615 


Let ex be the ground line ; Pp’ the centre of the base t’n’q 
on the horizontal plane ; Pp” the vertical projection of the ver- 
tex of the cone, @ any given point in the circumference of the 
base, and rq’, pL the horizontal and vertical projections of 
the slant side passing through q’; also, let ee’, er” be the ho- 
rizontal and vertical traces of the given plane. | 

Since the slant side is given by its horizontal and vertical 
projections r’q’, P’L, and the plane by its traces, we have only 
to find the projections of the required point by prob. 17, chap. 
2. The operation is as follows: Produce q’x' toc’ and x ; 
draw c’v and ur” at right angles to ea; join vr’ cutting P’L in 
s”, and draw s’s’ at right angles to ex cutting Ex and e’@’ in w 
and s’, and ws’, ws” will be the horizontal and vertical projec- 
tions of the point required. 
~ Ina similar manner we find the horizontal and vertical pro- 
jections. p’ and p” of the point in which the given plane is cut 
by the slant side oS passes through ® the other extremity 
of the diameter q 

If the point £ re at an infinite distance, the traces re’ and 
EF’ become parallel to the ground line ; and this circumstance 
produces a variation in the method of construction for some 
points that may require farther illustration. 





616 | DESCRIPTIVE GEOMETRY. 


Let as be the ground line; K@’c'R’ the base of the cone of 
the horizontal plane, r’ its centre, kp” the vertical projection 
of the axis of the cone ; and er’, e’n” the traces of the given 
plane, which are parallel to as. 

Suppose the horizontal projection qr’ of two opposite slant 
sides to be the diameter of the base parallel to an; and there- 
fore p’L, Pm the vertical projections of those sides. To de- 
termine the points in which the plane meets those slant sides 
we may proceed as follows : 

Make xp equal to Ko, join cD cutting qk’ in z; make xy" 
is to Pz, and through y” draw s’r" parallel to AB; ; then 

s's’ and 7’ being drawn perpendicular to as, will give the 
horizontal projections of the required points ; ‘and s’, T” the 
corresponding vertical projections. 

Again, to determine the intersection of the plane and slant 
side passing through x: make Pv’ equal to Ke" which is the 
altitude of the cone ; draw KV’ intersecting c’D in w’'; make 

wx parallel to as, and x will be the horizontal projection of 
the required point, and x’w will be equal to the vertical or- 
dinate, the horizontal ordinate being xx . 


PROBLEM III. . 


Lo construct the horizontal projection of the curve made by the 
entersection of a given plane wih a given conic surface. 


Let aB be the ground line; xe'c’ the circumference of the 
base on the horizontal plane, touching the ground line in x ; 
let p’ be the centre of the base, and at the same time the he- 
rizontal projection of the axis, and vertex ; and xp” the ver- 
tical projection of the axis. 

Suppose the given intersecting plane to be parallel to the 
ground line, or which is the same in effect, let the horizontal 
and vertical traces a’v’, T's” of the given plane to be parallel 
to aB, and suppose the horizontal trace a’v’ to touch the base 
of the cone in ¢’. 

To find the axis of the projection, make xa equal to kK” ’ 
and join ca in PRO’ paraltel to aB; take Po equal to KP’; 
join ko’ intersecting ac’ inv’, and from vu’ draw v'p’ parallel to 
4B, andc’n’ will be the axis of the projection. 

Again, to find the points in which the curve to be projected 
cuts the diameter @’r’ parallel to AB from @’ and Rr’ draw @’L, 
R'm perpendicular to az, and Px, pm will be the projections 
of the slant sides passing through q ‘and r/: let c’a meet P’o’ in 
rn’, and nee made ry’ “equal PR, draw H’y's” parallel to as, 
and yy, H’H' parallel to P”’p ‘and sx’ will be the points re 


quired i in QR. 
To 


DESCRIPTIVE GEOMETRY. 617 


To find the point in which the curve meets any other radi- 
us p's’, draw s'n at right angles to as ; join P’N which is the 
vertical projection of the slant side passing through s': pro- 
duce p's’ to meet a’v' and aB in v’ and 7; draw vB, tr” pa- 
rallel to P’p", and join et’, cutting e’N ine”; draw ee’ paral- 
lel to P’p and x’ will be the point in which p's’ is intersected 


by the curve. 


#h 4 . T VA Fr B” 


Te 4 pe 





a 








AN mie rr 


+ 


In a similar manner we may find any number of points in 
the required section c'e'DF’. 

When the points v, and r become to remote too be conse- 
quently used in the construction, we may find the required 
points of the curve by the method used in determining the 
intersection of the plane by the slant side passing through x. 

In this example, in which the cone is divided by the plane 
into upper and under parts of the conic section, is called an 


V OL.-dh: 79 ellipse ; 


618 DESCRIPTIVE GEOMETRY. 
ellipse ; the curve cepF is therefore the projection of an 
ellipse, ‘and consequently, c'e'p'F’ is also an ellipse. . 


PROBLEM IY. 


To construct the ellipse of which the curve c'e'v'r’ in the preced- 
mg problem ts the horizontal. 


This problem is readily solved by finding by prob.12 
chap. 01. the positions of the poinis of the curve on the hori- 
zontal plane by the revolution of the intersecting plane to a 
coincidence with the horizontal plane. 





Or, we may proceed as follows, which is nearly equivalent. 
Because xo’ is the position on the horizontal plane of the 
slant side passing through m, and KC'A the angle of elevation. 
of the given plane ; it is plain, that cv’ is the transverse axis 

ot 


DESCRIPTIVE GEOMETRY. 619 


of the ellipse section required. Make c’d equal to cv’, and 3 
will be the position on the horizontal plane of the vertex ile- 
termined by v’. 

In like manner, take c’# equal to cr’, and make the per- 
pendicular ay equal to p’p’, and » will be a point in the requir- 
ed section. In a similar manner, we may find any number of 
points in the circumference of the required ellipse dxc’@. 

It is evident, from this construction, that the ellipse dzc’p 
is derived from c'e’v'r’ be elongating each abscissa from ic as 
cp’ in the constant ratio of c’p’ to c’v’; so that cr’ is to co! as 

‘p’ to c’é, while the semiordinate am remains the same as 
PHY, 

And as the curve drc’¢ is by the definitions of conic sec- 
tions an ellipse, it is manifest from the cowstant ratio of the 
abscissas cp: and cr having a common semiordinate rH’ or 
#7, that the projection cw’ pF’ is also an ellipse. 


PROBLEM V. 


To construct the section of a cone by a plane parallel to the axis 
of the cone. 


Let ap be the ground line; r’the centre of the circular base 
xq'R of the cone touching the ground line aB in K: preduce 
the radius r’x to P’, and take xp” equal to the axis of the cone 
which is supposed to be at right angles to the plane of its base, 
and consequently to the horizontal plane ; then P’x is the ver- 
tical projection of the axis, and p” of the vertex. 

Suppose the cutting plane to be parallel to the vertical plane, 
and to intersect the horizontal plane in the straight line r’yq’, 
which is therefore parallel to a8, and consequently perpendi- 
eular to the diameter ck. 

Draw any radius P's’? of the base, meeting r’@’ im 8”, and 
the circumference of the base in T. Find by prob. }, ont 
tv, the vertical projection P’’v of the slant side passing through 
r, the corresponding horizontal projection of this slant side 
being e't : through s draws's’ at right angles to as, and meet- 
ing vp” In s", a ns’ is the altitude of the conic surface at 8’, 
because ns, ns” are evidently co-ordinates of a point of the 
slant side peso through r. 

And, since the cutting plane which passes through RQ’ it 
perpendicular to the horizontal plane, it is evident that ws’, 
ns’ are the: co-ordinates of the point in which the slant side 

terminating 


620 DESCRIPTIVE GEOMETRY. 


terminating in T penetrates the cutting plane ; if therefore we 
make s'x equal to ns”, it is plain that x will be the position on 
the horizontal plane of the point denoted by s’, s”, by the re- 
volution of the cutting plane about the intersection R'@’. 


BY 





By a similar construction, we may determine any number 
of points in the curve @’txr’, which will be the section re- 
quired. 

The curve required may be obtained still more simply by 
merely,finding the perpendiculars ns’H,and describing the curve 
through L,s",m, &c. without determining the corresponding 
points in @'LxR’. | , 

It is evident that the plane meeting the base at right angles. 
in P'y@’ mast also meet the upper division of the conic surface, 
and produce another section equal and similar to q’xr. The 
curve determined by this construction is an hyperbola. . 


DESCRIPTIVE GEOMETRY. 621 


~ PROBLEM VI. 


To construct the intersection of a conic surface by a plane pa- 
rallel to one of the slant sides of the cone. 


Let az be the ground line ; P’ the centre of the cone’s base, 
which is supposed to be coincident with the horizontal plane ; 
and let the base erx touch the ground line in x: in P’K pro- 
duced, take xP” equal to the altitude or axis of the cone, and 
p” is the vertical projection of the summit of the cone. Let 
the cutting plane be parallel to the ground line, and meet the 


G eae oe ices wala 





~ base in the horizontal trace EF, which will consequently be 
parallel to as : in xP’ produced if necessary, take xe” a fourth 


G22 ' DESCRIPTIVE GEOMETRY. 


proportional to the three straight lines xr’, kx, Ke”, and the 
straight line c’y’s” parallel to as, will be the vertical trace of 
the cutting plane. . 

The angle which the slant side passing through z makes with 
the horizontal plane is evidently the acute angie at the base of 
a right-angled plane triangle of which the base is zr’, and per- 
pendicular equal to KP’; and the angle which the cutting plane 
makes with the horizontal plane is also the acute angle at the 
base of a right-angled triangle of which the base is xx and 
perpendicular xy"; and since these two triangles are im the 
same plane and have the bases and altitudes proportionals, it 
is plain that the acute angles at their bases are equal, and that 
the slant side passing through z is parallel to the plane of 
which the traces are EF, 6H’. ne ! 

To construct the curve of intersection draw any radius F’e ; 
from @ draw Qo at right angles to ap, and join p’m, then Pa, 
and p’m are the horizontal and vertieal traces of the slant side 
passing through q. Find by prob. 2. chap. 1v the horizontal 
and vertical projections s’ and s” of the point in which this 
slant side meets the cutting plane ; and by prob. 12. chap. 11. 
find 1the position on the horizontal plane of the point of which 
sand s” are the projections by the rotation of the cutting plane 
about the intersection er, andi is a point in the required 
curve. 

In a similar manner we may proceed in determining any 
number of points in the required curve FIVE. 

The ordinates no, uw are obtained by the construction given | 
in prob 2, chap. rv. for the slant sides passing through the ex- 
treinities of the diameter pr parallel to the ground line az. 

The vertex v is found by taking ka equal to xy”, and r'p 
equal to xr’; then drawing ax and Kp, we have the position 
c of the vertex of the curve on the horizontal plane ; and 
therefore making xv equal to xe, the point v will be the ver- 
tex of the curve. 

It is obvious that the curve Five is a parabola. 


LOGARITHMS 








2 
3 |0 477121 
4 10.602060 
5 |0.698970 
6 0.78181] 
7 |0.845098 
8 lo.903090 





seta a 
0 | 1. an 
12 1.079181 
i .113943 








14 11.146128 
11.176091 
- 1.204120 
17 {1.280449 
18 1.255278 
1.278754 
2 0 |. 301050 
ain ,322219 
|1,342423 
37 rh etadl ec 
| 2411 3802113 


25 |t. "3979401 













































































OF THE 
NUMBERS 
FROM 
1 to 1000. 

N.| Log. i Leg. Ral ~ Log. 
26 |1.414973!| 51 |1.7075701| 76 |1.880814 
27 |1.481364 ate 716003!| 77 |1.886491 
28 |1.447158\| 53 11.724276|| 7g 11.892095 
29 |1.462398]| 54 |1.732394)| 79 | 897627 
30 |1.4771211| 55 |1.740363 1.903090 
81 |1.491362\| 56 17481881) 81 1.908485 
32 |1.5051501| 57 11.755875 1.913814 
33 onal 58 |1.763428. ns 1.919078 
1.531479]1 59 1,770852| 84 11924979 
5 |is4aossl 60 |1:778151| 85 |1,920419 
1.556303)! 61 1.785330) 36 {1.934498 
1 .563202'| 62 1'792399]| 87 |1.939519 
cd oeaannal 63 |1.799341|1 88 11.944483 
39 11.591065 64 [1.808180 8Y¥ 11.949390 
1.602060]! 65 |1.812913]| 90 [1.954243 
ai i-si2rse| 66 6 [1.819544 91 11.959041 
. 1,623249!! 67 fea 92 [1.963785 
3 |1.633468 cs | 832509! 93 11.968483 
44 |1,643453 i 1.838849 [94 | 973128 
4s [1.055213 70 | 835008 9 1.977724 
46 |}. 662758 851258! 6 1 982271 
aa) 672098 ee oeeeoul sb 1.986772 
ae 68124111 73 |1.863323]|| 96 |1. 991226 
49 690196|| 74 11.8692321| 99 11.995635 
0 |1,698970!| 75 11.87 875061||100 |2.000000 





N. B. In the following table, in the last nine columns of each 
page, where the first or leading figures change from 9’s to 
0’s, points or dots are now introduced instead of the 0’s 
through the rest of the line, to catch the eye, and to indi- 
cate that from thence the corresponding natural number in 

the first column stands in the next lower line, and its an- 
nexed first two figures of the Logarithms in the second 


ceoluma. 


'N.; O 1 


100|006000|0434/0868} 1301}1734| 216612598 
4321|4730|5181/560916038' 6466/6894 
8600/9026 9451/9876}. 


101 
102; 
103/012837/3259 
104, 7033/7451 


105/021189}1603, 


106} 5306/5715 
107| 9384/9789). 
108/033424/3826' 
109} 7426/7825 


110/041393|1787, 
111 
112) 9218/9606 
113/053078/3463 
114) 6905/7286 
1151060698|1075 
4458/4832 
8186/8557 
1118/07 1882|2250 





. 3943/4277 
727 1|7603 
120574|0903 





612816438 
921919597 





146] 4353/4650 
147| 731717613 


iol sisdsdral 


5323/5714 





3852/4178 


LOGARITHMS. 
5 4 





2 6 7 8 


302913461\3891 
7321|7748/8174 


4 



















3001. 72411147 
3680/4100}4521| 4940/5360. 
7868|8284187001911619532/994 
20 16|2428|28.41/3252|3664 
6125/6533] 694217350|7757 | 
', 195]. 6001100.4] 1408] | 812/2216/2619/3021 
4.227|4628|5029|5430|5830,6230| 6629/7028 
8223/862019017/94141981 1). 207]. 602). 998 
21821257 6|2969| 3362/37 55|4.148|454014932 
6105|649516885|7275|7664:8053|8442' 8830] 
9993]. 380}, 766|1153]1538/1924|2309|2694 
'3846|423014613/4996 5378|57 60 614216524 
7666|8046|8426)8805|9185/9563|9942|, 320 
1452|182912906/2582|2958'333313709 4083 
5206|5580|5953|6326|6699|707117443/7815 
8928/9298/9668). . 38]. 407|. 776|1145|151 
2617|298513352/3718|4085|445 148165182 











136! 3539/3858 
137], 6721|7037 
113 9879|. 194), 


5336|5640 
8362/8664 





6748/ 7058173671767 6|7985|8294/8603/891 1 
9835|. 142]. 449]. 756}1063} 1370|1676)1982 
2900!3205135 10}381514! 201442414728/5032 
5943|6246/6549]6852|7154|7457/7759|8061 
8965/9266|9567'9868]. 168]. 469]. 769|/1068 
1967\2266/2564/2863/3161/3461 1375814055 
4947,5244:5541158381613416430|6726}7022 
7908182031849 713792|9086/9380|9674|9968 





0848}1141]143411724)2019/2311|2603/2895 








_OF NUMBERS. 
[N.[ 0 1 |] 2 Ee A BEG 
1501 17609 1163811667016959|7248|7536 7825/811318401/8689 
1511 897719264/9552/9839|. 126], 413!. 699', 980]197911558 


152) 181844/2129)2415)2700)2985/3270) 3555 3839)4123|4407 
153} 4691]49751525915542)5825/6108)6391 6674 6956 he 











































5127|5347|5567, 
5|6884|7 104|7323|7542|776117979|8198|844 6/8635 
199] 83853/9071]9289/950719725|99431. 





197; 4466 4687 4907 





LOGARITHMS 




















PRO PO eS ia | sb ey 7 ee Lae 

200|301030]1247/1464/1681/1898/2114/2331|2547|/27 64/2980 
201] 3196/3412)3628)3844|4059 | 4275|449 1/4706)4921|5136} 
202 535115566|5781|5996|621116425/6639|6854| 70687282 
203} 7496/77 10|7924/8137|8351/8564|877818991/9204/9417 
204) 9630/9843 268}, 481]. 693]. 906]1118)1330|)1542 


206} $867|4078/4289|4499)47 10|4920|5130)5340)555 1/5760 
207| 5970|618016390/6599| 6809/7018|7227|7436|7646|7854 
208} 8063)/8272|8481/8689)8898'9 106/93 1419522)/97 30/9938 
209|320146)0354/0562)0769|0977/1184)139111598|1805|2012 
210}.. 221912426/2653/2839/| 3046132 52|3458|3665|387 1|/4077 
211}. 4282)4488/4694/4899) 5 105)5310)5516|57211/5926|6131 
212) 6336/6541/6745|695017155 #359 7563/7767 |7972|8176 


216 4454 4655 4856|5057|5257|5458)5658|5859|6059 6260 
217 Page BOnG 6860\7060 ene 7459|7659|7858|8058|8257 
218 


223| 8305/8500/8694/888919083/9278/9472|9666|9860)|.. 54 
22413502481044210636)0829|1023/1216|141011603}1796) 1989 
225) 218312375|2568)/2761|2954|/3147|3339|3532|3724)|39 16 
226] 4108/4301/4493/4685|4876|5068/5260|5452|5643)| 5834 
227|  6026/6217|6408/6599|6790|6981\717217366|7554|7744 
228] 7935/8125|8316 8506|/8696|888619076|9260|9456|9646 
229] 9835]... 25|.215!. 404). 593]. 7831. 97211161|1350|1539 
!2301361728|1917|2105 048|3236|3424 





2912/309 6/3280) 3464|3647|3831/4015|4198 4382 4565 
47 48|4932|5115/5298)548115664/5846|602916212'6394 
6577/67 59|6942|7124|7306/7488|7 670 7852/8034) 8216 
839818 580|8761/8943/9 124|9306|9487 |9668)9849|, . SO) 








240|38021 110392 1887) 
241} 2017/2197|237712557|2737 129 171309713277 (345613636 
242) 3815|3995/4174/4353|4533|4712/489 115070|5249|5428 
24 $606)5785)5964/6142|6321/6499|6677|685617034|7212 
244) 739017568/774617923|8101/827918456|86341881 1/8989 
245) 9166)9343/9520/969819875 . 2281. 405]. 582). 759 


46|390935} 111211288 1464/1641 1817 1993!2169/2345!2521 
247| 2697\2873|3048/3224| 34001357 51375 i1}3926/4101|4277|, 

. aa 44 52)4627 4802|4977 5152|5326|5501/5676 5880\6025 
249 6199 6374165486722 6896 7071 7245 7419 1759217766 | 











OF NUMBERS. 





0 Se or6 7 
250/39794018114 8287 846118634 8808 898119134 932819501 
9674/9847 192]. 365]. 538.711}. 883/1056)1228 














4845|4997|5150|5302|5454)5606 5758/59 10| 6062/6214] 
6366| 65181667. |682116973/7 125/727617 42817579] 7731 
7882) 8033/8184|8336/8487|863818789|8940|909 1/9242 
9392/954319694/984519995|. 146|, 296). 447 48 
289/460898/1048/1198/1348/1499]1649/1799/1948 
2398 2548/2697 |2847|2997|3146|/3296|/3445 
3893140421419 1/4340!4490/463914788)4936 
5383) 5532/5680/5829/5977|16126'627 416423) 657 1/6719 
6868701 617 16417 312|746017 608) 7756/7904 et i 
- 8847 8495|864318790|8938/9085,9233/9380/9527/9675 
9822/9969]. 116}. 263]. 410]. 557]. 704). 851/. 98/1145 
296'471292/1488/1585|1732) 1878\2025 2171 2318/2464/2610], 
297| 275612903|3049|319513341(3487|3633/3779|3925}407 1 
298]  4216)436214508/4653/4799/494.4)5090/5235/538 1/5526 
2991 567 1158161596216 107 6252 6397|6542'6687 16832 6976) 




















LOGARITHMS 
a AS a Ea Ma ae 


| | | 


300\4771211726617411|7555|770017 84417989 
$01! 85661871118855/8999|9 1431928719481 
3021480007101 51/029 4,04381058210725|0869|101211156}1299 
303} 1443/158611729]1872|2016|215912302;2445|2588/2731 
3041 2874/3016'3159)3302\8445|3587|3730|3872|4015/4157 
305| 4300!4442/4585|472714869|5011/515315295|5437/5579 
806|  57211/5863/6005161 47/6289 |6430|6572\6714|6855| 6997 
$07| 7138\7280\742117563'770417845|7986|8127|8269/8410 
308] 8551/8692/8833)897419 11419255|9396|9537|9677|9818 
309] 9958]. 299). 239!. $80]. 52U!, 6611. 801]; 941/108 1, 1222 
310149 1362)1502|1642)1782/1922/2062|2201|2341/2481|2621) 
311} 2760|2900/3040 erat $458|3597|373713876|4015 





7 8 9 


8133/8278) 8422 
9575|9719|9862 














312] 4155)429414433/4572/4711148501498915 | 28|5267| 5406 
318]  554415683'582215960/609916238|6376|6515|6653|679 1! 
$14] 6930/7068'7206|73441748317621|7 75917897 |803518173 
315] 8311|8448|858618724'8862|8999|9137|9275/9412|/9550 
316] © 9687|9824/9962|, . 99], 236]. 374]. 5111. 648]. 785}. 922 
3171501059] 1196|1333) 1470] 1607] 1744) 1880/2017 |2154)2291 
318| 2427/2564/2700/2837\2973|3109|324613382/35181S655 
319} 8791/8927|4063/4199|4335) 447 1|460714745|1878/5014 
$20} 5150/528615421/5557|5693 
821] 6505|6640|6776/69111704617181)7516|74511758617721 
822] 78561799118 126182601839 5|18530|8664/8799|8934/9068 
$23] 9203/9337|947119606\9740|9874]...9|. 143]. 277]. 411 
324151054510679/081310947/1081| 121511349]1482|1616|1750 
325). 1883/2017/215 112284/2418)25511268412818|2951}3084 
826} $218/3351|3484136)7/37 5013: 
327}. 4548/4681)4813)4946'5079|521115344|5476|5609/5741 
328]  5874/6006,6139|627 11640316535|6668|6800169 32/7064 
329} 7196)7328)7 46017 5921772417 85517987(81 191825118382 
330} 85) 4/8646)8777/8909/9040/9171|9303|9434/9566|9697 

88 9828/9959}. . 90}. 221]. 3531. 4841. 615]. 745]. 876]1007 

332/521138}1269}1400) 1530; 1661] 1792}1922/2053)2183/2314 

333] 244412575/270512835|2966| 309 613226'3356/5486| 361 6 

334| 874613876|4006/4136/426614396|4526|45656/4785|49 15 

335) 5045|5174)53804|5434)5 563] 5693/5822/5951/6081/6210 

336} 6339|6469|6598)6727|6856|6985|311417243|7372/750! 

337) 7%630/7759|7888/8016)/8145|/8274/8402/8531/8660|8788 

338] 8917/9045'9174)9302/9430|9559|9687|9815/9943}. . 72 

$39|530200/0328|0456)0584,07 1210840|09.68! 1096|1223]1351 

340] 1479)1607/1734)1862! 1990}2117|224.5|237 2/2500]2627 

341] 2754|288213009|13136132641339113518|3645|377213899 
$342] .4026/4153)4280/4407|4534| 466114787149 14|504115167 
343}  $29415421|5547|5674)5 800] 5927|60531618016306|6432 
344) 6558/6685/681116937|7063/7 18917 315|74411756717693 
345} 781917945\807 1/8197/8322|8448|/857418699/88251895 1 
346} 9076|9202,9327|/9452\957819703/9829|9954|. . 79}. 204 
347 |540329/0455)0580|0705)083010955|1080]1205]1330]1454 
348! 1579)1704}1829/1953/2078|2203)2327|2452|257 6/2701 
349! 2825129501307 4]3199/332313447|357 113696|382013944 























= 


OF NUMBERS. 
Wipe [eo einai ps Pee ee 7 Te Ce 


TEE ES eae SR A UE ERE (re Ao EES, (Prk ee 
—— |__| — | ———_ —— 


350/544068]419214316)4440)4564 tasers hop 4936 5060|5183 
351) © 53071543115 55515678/5802|5925'6049/6172 6296, 6419 
352| 65431666616789|69 13/703617 15917 282|7405|7529 7652 
353| 777517898/8021|8144/8267|8389|85 12/8635 8758888), 
354] 9003/9 126|92491937 11949419 616|9739|9861/9984. 196 
355|550228|0351/04731059 5/0717 |0840/0962| 1084) 1206!1328 
356] 1450]15721}1694/1816)1938|2060,2181/2303|2425 2547 
357} 2668)2790|2911/3033/3155|3276 3398) 3519|3640,3762 
358] 38831400414126/4247/436814489 4610|4731|4852'4973 
1359] — §094/5215)5336)5457|5578/5699 5820/5940)606! 6182 
360] 630316423/654416664/67 8516905/7026)7 146 7267/7387 | 
361) 7507|7627|774817868|7988/8 108'8228| 8349/8469 8589 
362] 870918829}894819068/9 1188/9308 942819548 9667 9787 
3631 9907|. 26]. 146]. 2651. 385), 504). 6241. 7431. 863, 982 
3641561101)122111340]1459|1578/1698/1817|1936|2055.2173 
2293}2412|253 112650) 276912887)3006/3125/3244,3362 
348 113600)37 1818837/3955/4074/4192/4311)/442914548 
4666|4784|4903)/5021/5139/5257|537 6|549.4156 12)5730} 
5848|5966|608 4/6202 |6320)/6437|65 55|6673)679 1/6909 
7026|7 14417 26217379/17497|7614|773217 849|17967|8084 
8202/8319|843618554/867 118788|8905|9023/9 140 9257| 
937419491/9608|9725|988219959|. . 76]. 193}. 309]. 426 
570543|0660107 76/0893! 1010/1 126]/124311359|1476 1599) 
1709}1825|1942|2058/2174)229 1|2407/2523/2639|2755 | 
287 2|2988/3 104/32 20|3836/3452)3568/3684'3800/39 15. 
4031}4147]4263/4379/4494/46 10/4726/4841/4957)/5072/ 
5188 15303/5419|5534)565015765|5880/5996/61 11,6226! 
6341|6457|6572)|6687 (6802/69 17|7032\7 147|7262)7 377 | 
| 749217607|7722/7836|795 118066 8181'8295/8410)8525' 
 863918754/8868/8983/90971921219326,9441/9555 9669; 
9784|9898|.. 12]. 126). 2411. 355}. 469]. 583], 6971. 811; 
58092511039 1458) 1267 1381}1495]1608)1722)1836)1950, 
2063/2177|229 11240425 18)263.112745|285812972|3085: 
$199|331213426|3539|3652/3763/3879|3992\4105/4218 
433114444145 57) 4670|4783/4896|500915 122/5235|5348, 
546115574|5686)5799/59 12|60241613716250|636216475) _ 
6587|6700|68 1 2!6925|70371714917 26217 37417 48617599 
771117823|7935|8047/8160'827 2|8384184.9618608|8720 
8832/8944|9056)9 167|92791939 119503|9615|9726|9834 
9950}... 61]. 173). 284]. 396]. 507]. 6191..730], 8421. 953 
390/591065|1176|1287/1399}15101162111732]184311955/2066 
391} 2177|2288|2399/25 101262 1\2732|/2843)2954/3064|3175 
392] 328613397|3508)/3618/3729/3840/3950!406 11417 1142821 | 
393) 4393)4503|4614/4724| 4834/49 45)5055|5165'5276/5386 
394} 5496}5606|5717/5827|5937|6047| 6157/6267 1637716487 
395| 6597|6707/6817|6927|703717 146|7256|7 3667 476|7586| 
396] 7695!780517914'8024/8 13418243|8353|8462|8572|8681 
397} 8791/8900|9009/9 11919228|9337|944619556|9666|9774 
398] 9883/9992}. 101). 210}..319), 428]. 537]. 646). 755). 864 
$991600973|108211191/1299]1408) 1517'1625|1734]1843(1951) 





























4001602060 
314413 


412 


413}. 


414 
4145 
416 


447|620136/0140}0344/04481055 2/0656 


ST nn er er er re ET ET : = s eee x 
© 
io) 


418 
ALS 
420 
42 | 
422 





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630428105301063 1073310835109 36 


LOGARITHMS 
192 3 ed | BC ee ES ee 
2169/2277 |2286 2494 2603/27 11|281912928| 3036} 
2531336 113469/3573|3686}3794/39021401014118 
4226/4 334/4442[4 9 50/4658) 47 66/487 4/49 82| 5089/5197 
§805|5413|5521]5628/573615844/595 1/6059) 6! 66/6274 
638 1/6489/6596)6 04'681116919|7026)7 133/7241|7348 
745 5|7562|7669|7777|7884|799 1 |8098|8205/8312|8419 
8 526/8633/8740/8847/895419061)9 167/927 4/9381/9488| 
9594/9701/9808]/99 14)... 21], 128]. 234].°341]. 447). 554 


0 








172311 82911936/2042) 214812254 23601246612572|2678 
2784| 2890/2996] 3102/320713313'341913525| 363013736 
3842/3947 14053)415914264'43701447 51458 114686/4792 
4897|500315 108/5213/5319)5424/5529/5634/5740/5845 
595016055161601 626516370! 64761658 1/668616790/ 6895 
7000)7 105172 10}73.151742017525|7629|7734/7839| 7943 
8048/8 1531825713 362|/8466/857 1/867 6/8780! 888418989 
9293191981930219 406195} 1 9615/9719 982419928]... 32 
076010864|0068| 1072}. 
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42821438 5144881459 114695/4798'490 115004 5107|5210 
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808218185|8287 
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468167024110339|0431/052410617 


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68033610426|0517/0607/0698)0789 


OF NUMBERS. 




























































N:) 0 If | 20 te Sp 4 | Be) 6 ae 9 
45016532 13|3309|3405 saan 3694|3791'5888'3984/4080| 
451| 4177|427514369|44651456214658/4754148 50/4946|5042| 
452} 5138152355331 5427 5526 561915715158 101590616009] 
453} 6098|6194|6290/6386 6482/6577|6673|676916864| 6960) 
454) 7056|7152|7247|7343/7438/7534|76291772517820|79 16] 
455| 801118107/8202/8298 $393) 8488 8584) 8675 8774/8870) 
456] 8965}906019155'9250/9346/9441|/9436'963 1|9726|9821 | 
{457} 9916]. .11}. 106!. 201]. 296|. 391), 486|. 881}. 6761. 771 
458|66086510960| 105 5)1150|1245| 1339) 1434/1529|1623|1718) 
459| © 1813}1907/2002}2096/2191/2280/2380/2475|2569/2663. 
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461} 370113795}38891398314078/4172|4266|4360/4454/4548 
462)  4642|4736|4830/ 4924) 50181511215206/529915393|/5487| 
463| §58115675|5769/ 586215956 605U!6143|62371633 116424. 
464} 651816612|6705'6799 6892|6986 7079\7 173|7266|7360! 
4651 745317546|7640|7833'7826|7920180131810618199|82931 . 
466| 838618479|8572'8665 8759/9852 8945 2035/9131 9224) 
A67| 9317|9410|9503|959619689 a4 


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480| 12411133211422}1513/1603|/1698 (784 1874|1964'2055) 
481) 2145|22351232612416/2506/22.96|2636/2777/2867|2957 1 
482| 3047|3137|3227|3317|/3407|3497|3587|3677|3767|3857 | 
483) 3947|4037|4127/4217'4307 |4396/4486|457614666|4756| 
484} 4845149351502515114/520415294 pele 5563/5652) 
485| 5742|5831/5921|6010/6100/618916279|6368!6458|6547 | 
486| 6636|6726|6815|6904/6994 rosa}ri73(726 735117440} 
487| 7529)7618|7707|779617886'797518064|8155/8242/8331| 
488) ~ $420/850918598/8687|8776/8865/8953|9042 9131|9220! 
489| 930919398]9486,9575'9664,9753'9841/9930I.. 19). 107 
490|690196|098510373]036210550\0639|0728l0816 0905} area 
491} 1081|1170|1258/1347|1435/1524'1619/170011789;1877 
492} 1965/205512142/2230/2318|2406|2494/2583/267 1/2759 
493} 2887/2935)3023/3111'31991328713875/3463'3551/36394 
494} 3727|3815}3903|399 1|4078/4166|4254/4342 relat 
495} 460514693/478 1/4868 495615044151 31/521015307/639 
496) 5482/5569 5657/5744) 5832|5919|6007 Betas eh: 
497) 6356|5444)653i|6618|67061679316880/6968!7055)7 142 
498| 72291731717404 TAQ TSS 76657752 eta 8014 
1499] 8101]8188/8275'8362|8449185351862218709|8796 88e3| 





LOGARITHMS 
1 | 2 


















N.} 0 (Sila) | Steele S tap 
600177815 1|8224)829 6|8368/8441|8513|8585|6658/8730|8802| 
601] 887418947/9019]9091/9163/9236/9308|938019452/9524 










. 29). 101]. 173). 245 











~1037)1109)118111253)1324) 1396) 1468/1540) 1612/1684 







605). 175511827|18991197112042|2114/2 186/2258/2329|2401 
606| 3473/254412616|2688/2759}283112902|297413@46/3 147 
607| 318913260/33321340313475|354613618)/368913761/3832} 
608) 3904|3975|4046|4118/4189/426 11433214403/4475|4546 
609| 4617|468914760/483 114902) 497415045|5116|5187|5259 
610] 5330154011547 2)554315615| 5686157 57|5828|589915970 
611} 604116112|6183/6254|/6325|6396|6467/6538|6609|6680 
612! 6751|682216893169641703517106/7177|7248|7319|7390 
613] 7460|7531|7602|7 67 317744|7815|7885|7956|8027|8098 
614! 8168/8239/8310/838 11845 1|8522|8593|8663|8734\8804 
615| 8875|8946|9016|908719157/9228/9299|936919440|95 10 
616} 9581/9651/972219792)9863|9933|...4).. 74). 1441.25 
6171790285|0356|0426|049 6/0867 |063710707|07 78/0848 |09 18 
618} 0988/1059]1129]1199} 1269) 1340]1410)1480/1550}1620 
619] 1691}1761/1831]1901/1971/2041/2111/218 1/2252/2322 
620]  239212462'2532/2602/2672/2742/28 1 2/2882/2952/3022 
621} 309213162/3231133011337113441/451 1/3531/3651/3721 
622| 3790/3860|3930)4000)/4670/4139/420914279]4349/4418 
623] 4488/45358)4627|4697|4767 |4836/4906|497 6|5045|5115} 
624] 5185|525415324/539315 463) 5532156021567 2/5741/581 3 
625| 5880/5949\6019|6088|/6158\6227|62971636616436|/6505 
626 


6574|6644167 13/6782) 6852/6921 699017060/7 12917198 
7268)7337|7406|747 51754517 61417683)/7752/7821|7890 





628} 7960/8029| 8098/81 67/8236 8305)/837 4|8443/8513/8582) 
629} 8651/8720/8789|8858/8927 8996|9065|9134/9203/9272 
630} .9341]940919478|9547/9610,9685|9754)9823/9892|9961 
631/800029|0098)0167/0236/0305 0373/0442/0511/0580)0648 
632} | 0717|0786|0854/0923 0992 1061/1129]1198|1266/1335 
633] 1404/1472|1541}1609|1678: 1747) 1815)1884}1952)/2021 
634} 2089}2158/2226|2295/2363, 2332/2500|2568|2637/2705} 
635}  2774/2842/29 10/2979|3047 3116/5184|3252/332 1/3389 
636} 3457/3525/3594|3662/3730 3798|3867|3935|4003/407 t 
637| 4139/4208/4276|4354/4412'4480/4548]/4616/4685)4753 
/638| 4821/4889)/4957|5025|5093/5161)5229/5297|5365/5435 
639]  550115669|5637|5705|5773/5841/5908/5976|6044/6112 
(640 6180/\6248)63 16/6384 64516519 6587|5655\6723|6790. 
1641) 6858)6926/6994/7061|7 129.7157 |7264/7332/7400/7467 
(642) 7535\7 603|7670)}7738/7 806 7873|7941|8008/8076/8143 | 
643} 8211)827918346/8414/848 1 8549/8616)8684/875 1/8818) | 
644| ~ 8886/8953|9021/9088|9 156,9223/9290|9358)/9425|9492 
645] 9560/962719694|97 62/9829 9896/9964). . 31}... 98). 165 


| nee 0300/0367 !0434/0501, '0596/0636(0703|0770|0837 

647| 0904109711039} 110611173/1240/1307|1374) 1441/1508 
red 1575) 1642/1709! 17:76|184311910/1977/2044/2111/2178 
1649} 224512312 2379/2445 2512|257912646|2713/2780'2847 


——— | 


650/8 12913/2980/3047 


OF NUMBERS. 
coe Es ae 6p 718 | a: 
3114|3181|3947|3314|3381|3448/3514 

















































































































651 ‘et hep 37 14/378 1)3848/3914'3981/4048/4114/4181 
652] 4248)4314)4381/4447|45 14/4581/4647/47 14|4780/4847 
653| 49 ty 504615113/517915246/5312/5178/5445/5511 
654] 5578/5644)57111.777|5843|59 10} 597 6/6042/6109|6175 
655} 6241/6308/637416440: sed el 6639|6705|677 116838, 
656] 6904/697017036|7 i02/7169|7235|7301/7367|7433|7499 
657} 7565)7631 i 7764|7830|7896|7962|8028}8094/8160 
658} $226/8292 ee 8424|849018556/8622|8688187 54/8820 
659} §8885/8951/9017/9083/9149|9215)9281 shy 9412/9478 
660} 9544/9610/9676/9741/9807|987 3/9939]. .. Ol. 136) 
661139020110267/083310399|0464/0530/0595 nee 0727/0792 
662}. 0858/0924/0989/1055/1120/1186/1251/1317}1382)1448 
663) 1514/1579}1645 ee 1841] 190611972/2037|2103 
664] © 2168/2233/2299|2364 24302495 2.560|2626]269 1/2756 
665! 2822/2887) 2952)3018/3083/3148]3213/32791334413409 
666) 3474/353913605|3670 3735|3800 3865/393013996|4061]. 
667| 4126|4191/4256/4321/4386/4451/45 16/458 11464614711 
668} 4776/484114906/4971 5036|5101 5166|523115296|5361 
669} §426|54.9 1/5556) 562 1/5686/5751)/58 1 5/5880/5945)6010 
670 aoe eleias 6204 6269 6334/6399 6464)6528] 6593/6658 
671) 6723|6787|6852|6917|6981|7046/7111|7175|7240|7305 
672) 7369|7434 7499)7563 7628|7692/7757/7821|7886|7951 
673!  8015/8080\8144!820918273/8338) 8402/8467 | 853 118595 
G74) 8660/872418789|88531891818982/904619111]917519239 
675] 9304/9368 epithe 9561/9625|9690/9754|9818/9882 
\676| 9947|.. 11}. - 139}. 204 268}. 332]. 396]. 460}. 525 
6771830589 0653 071 ; 0781/0845|0909|0973|1037|1102|1166 
678 ne Tee 161411678] 1742] 1806 
679} 1870}1934!1998!206212126/2189/2253/2317/238 112445 
680 2509/2573 2637 2700|976s|2828 2892/2956] 3020/3083 
681} 3147 ae 327513338 yal 346613530/359213657/372) 
682} 3784/384813912/3975 4039| 4103|4166|4230| 4294/4357 
683)  442114484'45481461 1!467514739|4802'|4866/4929/4993 
684 5056| ls120]183 6247|6910|5973 5437|5500| 5564/5627 
685) 5691/5754'5817'588 1!5944,6007|607 116134/6197|6261 
686] 6524 16387|6451 6514|657 66416704|6767|6830|6894 
687} ° 6957) 7020!708317146'7210!7273/7336|739917462|7525 
688 7588|7652| 77151777 81784117904|7967|8036!809 3/8156 
689) §8219)8282'8345!8408'847 1/858418597|8660|8723/8786 
690 2849|8912]s975/9039 910119164]9227/9289|9352|9415 
(|691| 9478)9541'9604 fae 9729|9792|9855.9918|9981)|.. 43} 
692 840106|0169 seo 0357104201048 2,0545/0608|067 1 
693} 0733)/07961085910921!0984! 164611109 1172]1234/1297 
694} 1359)1422 1489] 1647 1610 167211735 1797] 1860/1922 
695). 1985)2047:21 101217212235!2297/2360 2422/2484/2547 
696] 2609 sine [2859 2921|2983)3046| 1108/3170 
697|  3233)329513357!3420 3482!3544/3606/36691373 113793 
698| 385513918 ye [4104/4166 42291429 114353144151 
699| 4477|453914601/466414726147881485U/49 12/4974] 5036 











ap aaa cae a se la ic Bi i oe ie Li aL ad 


LOGARITHMS 
182 [ae 















5718/5780/5842 
6337/6399 6461|6523 658516646 
--6955170171707917141 1720217264 
7573176341767 617758{7819|783 1 
8189/8251|8312|8374|8435|8497 
880518866189 28] 89891905119 112 
9419]9481/954219604/9665'9726 
850033/0095|01561021710279/0340 


5904}5966/6028 


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6090/6151/6213)6275) 
6708; 6770/68 32)/6894 
7326|7388}7449/7511 





7943/8004 
8559/8620 
9174)9235 
9788/9849 
0401|0462 





8066/81 28} 
8682)8743 
9297|9358 
9911/9972 


0524/0585) _ 









0646|0707/0769|}083010891/0952| 1014/1075 
1258)1320)1381/1442/1503)/1564! 1625/1686 
1870\1931}1992/3053/2114)2175' 2236/2297 
2480|2 5411260219663] 2724/2785/2846/2907 
3090/3 150/321 11327213333/3394'13455|3516 
3698/37 59|3820/388 11394 1/4002|4063|/4124/4185/4245 
4306}4367|4428] 44881 4549/4610] 467014731/479214852 
4913/497415034]509515156/5216|527715337 1539815459 
5519|5580/5640]5701| 5761/5822) 58821594316003| 60641 
6366|6427164871654816608|6668 
6970/7031)709 317 1521721217272] | 
757 4\7634|7694/7755|7815/7875 
8176/8236, 8297/8357/8417|8477 
8778' 8838] 8898/8958/9018/9078 
9379)943919499|9559|9619/9679 
19978]. . 38]. . 98]. 158], 218]. 278 
057 8|0637|069710757|0817|/0877 
1176/1236 1295/1355/1415/1475 
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2966/3025! 3085|/314.4'3904|3263 
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5933/5992) 6051/6110'6169/6228 
6524/6583) 6642) 67.01/67 60/68 19 
711417 173|72321729117350|7409} 
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-  $90213960)4018) 4076] 4) 34/4192]42501430814360/4424 
91 448214540]45981465614714/477214830]/4888l4945'5003 
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2968/3029 
3577|36387 









































1573} 1'631}1690]1748 
2156/2215|2273|/2331 
9739/2797/2855|2913 











OF NUMBERS. 





751} 5640/5698/5756 38135871 5929|5987|6045|6102|6160 
752|  6218/6276|6333|639 1|6449/6507 |6564|6622|6680|6737 
753) 6795/68 53/6910|6968/7026|7083|7 141|7199|7256/7314 
754| 7371/7429|7 48717 544|760217659|7717|777 4178327889 
755] 7947/8004|8062/81 19/8177 |8234/8292|8349|8407 |8464 
756]  8522'857918637|8694|8752|8809|8866|8924|898 1/9039 
757|. 9096,9153/921 1/9268/9325|9383|944019497/9555/9612 
758| 9669|9726/9784'9841)9898/9956|.. 13]... 70}, 127}, 185 
7 59|880242\0299]0356|U413/047110528|0585|0642/0699|0756 
760 
761 
762 


08141087 110928/0985/1042/1099}1156)1213 
1385) 1442)1499/1556)1613)167011727|1784 
1955)/2012/2069/2126/2183/2240)2297/2354/2 


124 !|1328 
1841/1898 
411)2468 


763 
764 
765 
766 
767 
768 


2525 
3093 
366) 

4229 
A795 
5361 


258112638) 262 5127 5212809/2866|2923 
315013207\3264/332113377|3434)349 1 
37 181377513832 )388813945|4002/4059 
4.285|4342/4399/4455/451214569/4625 
4852|4909|4965|5022|507815135|5192 
§418)547415531|558715644|5700|5757 





29 80|3037 
3548/3605 
411514172 
4682/4739 
5248|5305 
5813/5870 


769 
770 
77] 
772 
773 
774 


5926|5983160391609616152| 6209/6265 
649 116547|6604) 6660/67 1 616773}6829 
7054711117 167|723317 280) 7336|7392 
7617/7674177301778617842)7898|7955 
8179|8236|8292/8348/8404|8460/85 16 
874118797\8853|890918965|9021|9077 
775| 930219358)94141947019526|9 58219638 
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2095) 2 150)/2206)2262)2317|2373)2429 














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1782] - $207|3262/3318)/3373|3429|/3484|3540! 3595|3651)3706| 
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784) 4316|437 1|4427|4482/4538|4593|4648/4704|47 59148 14) 
785| 4870/4925|4980/5036|509 1|5146|5201/5257/53 1215567 | 
786] 5423/5478/5533|5588/5644|569915754'5809/5864/5920 
787| 5975|6030|6085|6140/6195|6251|6306'6361/6416/6471 
788] 6526/1658 1|6636|6692|6747|6802|6857|69 12|6967|7022 
789] 7077(7.132/7 187|724217297|7352|7407|7 46217 517|7572 | 
790| .7627|7682|7737|779217 847|790217957|80! 218067/8 122 
791]  8176\8231/8286/834118396/8451/8506'8561/8615\8670. 
792| 8725|878018835/889018944|8999'9054'9109|9 164/92.18. 
793} 92731932819383|943719492 oat 9602'9656|9711|97.66 
794, 982119875|9930/99851, . 39]. 149].-203]. 258]. 312. 
795|900367|0422|0476|053 110586 eee 0695 0749/0804/0859 
796) 0913|0968|1022] 1077/1131} 1186/124011295|1349|1404 
797|  1458}1513|1567|1622/1676| 1731/1785 1840|1894|1948 
1798] 2003/2057 /2112/2166/2221 |2275|2329 2384124382492 





799 


2547|2601|265 5/27 10|/2764|2818!2873 2927/298 1|3036 


4 


- LOGARITHMS — 
a SO 6] 7), 8) 9] 


8001903090 3144 319913253|3307|3361/3416|34701352413578 












80] 3633 36871374113795|384913904'3958 4012}4066 4120 
802} 417414229|4283/4337143911444514499145531460714661 
803! 4716|4770|48241487814932/49 86|5040|509415 14815202! 
804) 52561531015364|5418|5472|5526|5580156341568815742 
805| 579615850/5904|595816012|6066/611916173|622716281) 
806] 6335|6389|644316497|655 1/6604/6658|/67 12|/6766/6820 
807| 6874169271698 117035170897 14317 196172501730417358 
808} 74111746517519\757317626|768017734177871784 117895 
809} 7949}8002/8056/811018163/8217|8270/83241837 818431] 


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9021|907 4/9128/91811923519289|1934219396|944919503 
9556/9610/9663197 16197 70198231987719930/9984). . 
319 10091/014410197\0251103041035810411/046410518|0571 
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1815}. 1215811211/126411317| 1371} 2424/1477] 153011584]1637 
1690| 1 743)1797|1850] 1903/1956)2009/2063/21 16/2169) 
222 2)227 $/2323}2381|2435/2488/2541|2594|2647|2700 
2753/2806|2859129 13|29661301913072|312513178/3231} 
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38 14/3867}3920|397 31402 6|407914132]4184/4237/4290) 
434 3)4396/4449|4502|4555|4608/4660/47 13/47 66/4819), 
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8555|8607|8659|87 12|8764188 1 6|8869|892118973/9026 
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| $321920123/01 7610228 .0280/033210384/0436|0489/0541/05931 
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1166}1218/1270!1322}1374|1426)1478/ 153611582 1634 
168611738 1790] 1842 1894/1946|199812050 



















2206/2258] 23 10'2362|2414/2466|/25 18|2570|2622|2674 
2725|2777|2829/288 | |2933}2985|3037 1308913 1 40/3192 
3244/329 6}3348133991345 113503)3555|3607 

3762|3814}3865/39 17/39 69/402 1/4072/4124/417 6|4228 

4279\433 1/4383]4434]4486/4538/4589|4641|4693/4744 
“4796 /4848}4899/4951|500315054/5106/5157|520915261 

5312/5364 541515467 5518/5570|5621|5673/5725|5776 
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1g46| %370|7422 7473\7524 75761762717678|77 30|7781|7832 

















lea7| 7883\7935|7986)8037|808 8/8 140/819 1/8242)829318345 
1948] 8396/8447|8498/8549| 8601/8652/8703/87 541880518857 
1849| 8908|8959/9010/906119112'9163|9215|9266 


9317/9368. 


OF NUMBERS. 
0 co aan mers, aon 4 5} 





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6514|6564|66 1 4) 6665/67 15/67 65/68! 5|/6865|69 1 6|6966 

7V17\7 167 7217/7267|7317|7367|7418\7 468 | 
7618|7668 7718|7769|7819|7869)/7919|7969. 
8119/8169 §219/8269| 8320|837018420/8470) 
8620/8670 8720/87 70|8820|8870|89 19|89701 
9120/9170 9220|9270/9320/93691941919469! 
9619/9669 97 19/9769|9819)9869/991819968 
0118/0168 0218)0267/0317/0367|0417|0467 | 
(06.16,0666 0716|0765/0815|0865/091510964| 
1114/1163) 1213}1263) 1313) 1362)1412]1462} 
1611|1660)1710)1760] 1809) 1859|190911958' 
2107/2157 |2207|2256/230612355|2405|2455: 
2603/2653 2702/27 52/2801128511/290112950 
3099|3!48 3198/324713297|3346|3396/3445 
3593/3643 3692/37421379 113841/3890|3939 
4088/4137 ,4186/4236/4285| 4335/4384/4433) 
4581|4631'4680/4729|4779| 4828) 4877/4927 
5074/5124/5173/5222 52729153211/5370|5419 
5567/5616) 5665/57 15|5764/5813/5862/5912| 
6059/6 1081615716207 |6256|6305|6354/6403 
6452|/6501/6551/6600/6649|6698|674716796|6845|6894 
69 4316992|7041/7090 7 140|7 189|7238/728717336|7385 
7434|7483}7532'758117630|767917728/ 7777/7826 7875) 
7924/7973) 8022|8070/81 19/8168)8217|826618315|8365 
8413/8462/8511/8560/8609|8657 8706 8755|8804|8353 
8902/895.1|8999/9048|9097|/9 14619 19519244/929219341: 
939019439 |9488'9536/9585|/9624)968319731/9780/9899 | 
9878|9926|9975|.. 24|.. 73). 1211. 170), 219]. 267). 316. 
0462/0511/0560|0608|0657/0706|0754/0803 | 
094910997] 104611095/1143/1192/1240)1289 
1435) 1483]1532)1580|1629|1677|1726|1775 
1920|1969/2017|2066) 21 14/2163/2211/2260. 
2405/2453/2502/2550/2599|2647;2696,2744 
| -2792)284112889|2938/2986/S034| 3083/3 13113180/3228 
327613325 |3373)/342113470)3.518|356613615|3663)37 11 
376013808 3856/3905)39531400114049!409814146,4194| 


a eee 


051610566 
1014/1064 
1511/1561 
2008/2058 
2504/2554 
3000) 3049 
349513544 
3989/4038 
4483/4532 
4976/5025 
 §46915518 
5961/6010 











085 !}}0900 
1338/1386 
1823/1872 
2308}2356 














LOGARITHMS 
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a mn es fe I ie! 


4435|4484|4552|4580|462814677\ 
4918)4966|5014|5062/5110/5158}- 
902! - 52071525515303/53a1 s390|5447 549515543/559215640| 
903| . §688/5736/5784/583215880/592815976|6024|607216120|- 
904} 6168/6216/626516313/636116409164571650516553/6601] 
905| 6649/669716745 679316840 688816936 sha phar 


—— | —_ | | - | 


900/954243/429 114339)4387 
901! 4725|4773)/4821 4869] 








906] 712817176)7224|727217320]7 36817416|74641751917559) 
907| 7607|7655|/77031775117799|7847|789417942/7990|8038 
908} 8086/8134/8181)/8229/8277/8325/8373/842118468/8516 
909} 8564/8612/8659/8707|8755)\8803/8850|8898/8946/8994| 
910}. 9041/9089/9137/9185)923219280/93281937519423194711 
911] . 9518/9566)9614/96611970919757/9804|9852/9900/9947 
912} 9995]... 42].. 90]..188). 185]. 233]. 280], 328]. 376]. 423 
9 F3'960471|051810566|0613/066 110709,0756|0804 rote 
914) 0946/0994 1041}1089 1136/1184)123111279|13826)1374 
915} 1421)1469}1516)1563/1611/1658]1706| 1753) 1801/1848 
916}. 1895|1943}1990 ee 2132/2180'2227|227512329 






















































917}  2369/2417124641251 1/2559/2606/2653/2701)}2748|2795 
918} 2843!2890/2937|2985|3032!307 9/3 126/3174/3221/3268. 
919] . 3216/3363/3410/3457|85041355213599|3646 oe 
920] 3788)383513882)392913977|4024/407 114118/4165/4212 
991| 4260'4307/4354/4401 448 4495|4542|4590 He beet 
992} 473114778]4825/487 2149 19|4966/5013/506115108/5155 
923) 5202/5249/5296/5343/5390!5427/5484|553115578|5625| 
924 5672/57 19 5766|5813/5860)5907|5954'6001/6048/6095 
925! 6142'6189/6236/6283/6329 8376) 64231647016517/6564 
926} 6611 8658/6705 6752/6799 68.45 689216939169 86/7033 
9927) 708017 127\71731722017 267 7314, 736117408 bepeies 
998} 7548/7595|7642 7688/7735 7782\7829/7875|7922|7969 | 
999} 8016/806218109/8156 8205/8249 ee 339018436 
930] 8483/8530/8576/8623' 8670/87 16|8763/8810]/885618903 | 
931} 8950/899619043/9090/9 pees: 9276|9323|9369 
932} 9416'9463]9509 eal ane 9649 9695197421979 9835] 
933] .9882|9928]9975). ate 161], 207). . 300[ 
9341970347/0393|0440 a ie 0579 062610673|071910765 


935| 0812/085810904/0951 Hebe lo 1090]1137,1183/1229 
936| 1276/13292/1369|1415114611150911554/1601 eric. 
937| 174011786|1832|1879/19251197}|/2018/2064)21 10]2157 
938] 2203/2249|2995|2342/238812434/2481/2527|2573/2619) 
939). 2666/27 12}2758/2804/2851/2897|29438 5|3082 

940} 3128/3174/3220/3266'3313/3359|3405'3451|3497'3543 



































941) 3590)3636}3682|3728)377413820\3866 iene 6k 4005 
942|. 405 1|4097|4143/4189/4235/428 1/4327! 437 4/4420)4466 
943| 4512)4558'4604/4650/4696147 42/4788] 4834,4880' 4926 
9441 4972/501815064/5110/5156/5202 mets 534015386 


945) 5$432)5476|5524'5570 


| 5616|5662/5707 5753/5799 5845). 
946] 58911593715983/6029 


6075|612 1/6167 6212/6258 6304; 











947| 6350/6396|6442/6488)6533)16579 6625,667 16717, 6763+ 
948}  6808|6854|6900]6946/6992)/703 7\7083|7 129) 717517220]. 
949! 7266|7312|7358/7 40317449 7495 7541,7586 763217678 








RDO EN BR Ss MNRAS, 


aa a RS 


1 2 


OF NUMBERS. 


3 


4 


5 


,—— | | |S ] |] 





iN, 0 
19501977724 
1951] 8181 
1952) 8637 
1953} 9093 
(954] 9548 
9551980003 
1956| 0458 
1957} 0912 
1958] 1366 
1959]. 1819 
1960| 2271 
1961 2723 
1962) 3175 
1963| 3626 
964) 4077 
1965) 4527 
1966| 4977 
967| 5426 
1968]. 5875 
1969] 6324 
1970 6772 
1971) 7219 
1972| 7666 
1973) 8113 
1974] 8559 
1975| 90805 
1976| 9450 
19771 9895 
19781990339 
(979| 0783 
1980| 1226 
1981} . 1669 
1982] “2111 
983| 2554 
1984] 2995 
'985| 3436 
1986| 3877 
1987) 4317 
9881. 4757 
1989! 5196 
\990| 5635 
991} 6074 
992| 6512 
9931 6949 
994| . 7386 
995| 7823 
996| 8259 
997| 8695 
998| 913] 
999] 9565 


Vou. Il, 


7769|7815 
8226 | 8272 
8683 8728 
9138'9184 
9594) 9639 
0049 | 0094 
0503/0549 
0957! 1003 
1411] 1456 
1864) 1909 
2316) 2362 
2769 2814 
2220 | 3265 
3671| 3716 
4122} 4167 
4572| 4617 
5022 | 5067 
547115516 
5920| 5965 
6369 | 6413 
6817| 6861 
72641 7309 
7711/7756 
8157| 8202 
8604 | 8648 
9049} 9049 
9494| 9539 
9939| 9983 
0383 | 0428 
0827 | 0871 
1270/1315 
171311758 
2156 | 2200 
2598 | 2642 
3039 | 3083 
3480| 3524 
3921| 3965 
4361) 4405 
4801) 4845 
5240 | 5284 
5679 | 5723 
6117/6161 
6555 | 6599 
6993 |7037 
7430 | 7474 
7867 | 7910 
8303 
8739 
9174 
9609 








7861 
8317 


8774. 
9230 


9685 
0140 
0594. 
1048 
i501 
1954 
2407 
2859 
3310 
3762 
A212 
4662 
5112 
5561 
6010 
6458 
6906 
7353 
7800 
8247 
8693 
9138 
9583 
28 
0472 
0916 
1359 
1802 
2244 
2686 
3127 
3568 
4009 
4449 
4889 
5328 
5767 
6205 
6643 
7080 
7517 


695) 


19739 


7906 
8363 
8819 
9275 
9730 
0185 
0640 
1093 
1547 
2000 
2452 
2904 
3356 


13807 


A257 
4707 
5157 
5606 
6055 
6503 


7398 
7845 
8291 
8737 
9183 
9628 
ihe 
0516 
09.60 
1403 
1846 
2288 
2730 
3172 
3613 
A053 
4493 
A933 
5372 
5811. 
6249 
6687 
7124 
7561 
7998 
8434 
8869 
9305 


C 


7952 
8409 
8865 
9321 
9776 
0231 
0685 
1139 
1592 
2045 
2497 
2949 
3401 


3852 


RS 


4502 
A752 
5202 
5651 
6100 
6548 
6996 
7443 
7890 
8336 
8782 
9227 
9672 
Fz 
0561 
1004 
14.48 
1890 
2333 
2774, 
3216 
3657 
4097 
4337 
4977 
5416 
5854 
6293 
6731 
7168 
7605 
8041 
8477 


8913 


9348 


9783 


ee 
feta barnes 
7998| 8043 
8454] 8500 
8911| 8956 
9366|9412 
9821|9867 
0276|0322 
0730]|0776 
1184] 1229 
1637/1683 
2090|2135 
2543|2588 
2994! 3040 


3446 | 3491 | 


3897 (3942 
ASAT; 4392 
AT97 A842 
5247 | 5292 
5699] 5741 


6144) 6189 


6593 | 6637 
704017085 
748817532 
793417979 
8381] 8425 
8826|8871 
927219316 
971719761 
. 1611. 206 
060510650 
1049}| 1098 


1492| 1536 


193511979 
2377) 2421 
281912863 
3260} 3304 
370113745 
A141] 4185 
458) |4625 
5021] 5065 
5460| 5504 
5898| 5942 
6337 | 6380 
6774| 6818 
721217255 


764817692 


8085] 8i29 
8521) 8564 
895619000 
939219435 
982619870 


i 
| | 


ne TN 
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LOG. SINES, TANGENTS, &e. 





0 Deg... . 1 Deg. : 
~ Cosine. Tang. Sine. | Cosine. { Tang. _ Cotang. _ 


1 10.000000 Re 8.241855| 9.999934] 8.241921|11,758079160 
6.465726 10.000000} 6.463726] 13.536274]| 8.249033] 9.999932) 8,249102|11.750898|59 
6.764756|10.000000} 6.764756] 15.235244|| 8.256094] 9.999929] 8.256165|11.743835|58 
6.940847} 10.000000} 6.940847] 13.059153|| 8.263042} 9 999927) 8.263115]11.736885|57 
7.065786|10,000000) 7.065786 8.269881} 9.999925) 8.269956/11.730044}56 
7.162696|10.000000| 7.162696] t2.837304|| 8.276114; 9.999922) 8.276691|11.723309|55 
7.241877| 9.999999) 7.241878|12.758122|| 8.285243) 9.999920] $.283323/11.716677|54 


7.3088241 9.999999) 7.308825) £2.691175|| 8.289773] 9.999918) 8.289856/11.710144153 
7.366816} 9.999999] 7.366817|12.633183)]| 8.296207} 9.999915} 8.296292/11.703708152) 
7.417968} 9.999999) 7.417976) 12.582030!| 8.302546) 9.999913] 8.302634]11.697366/51| 
7.463726} 9.999998) -7.463727|12.536273}} 8.308794| 9.999910} 8.308884/11.691116|50 
7.505118} 9.999998] 7.505120 3.314954] 9.999907] 8.315046|11.684954) 49! 
7.542906] 9.999997] 7.542909|12.457091]| 8.321027! 9.999905| 8.321122|11.678878|48} 


7.577668) 9.999997! 7.577672|12.422328|| 8.327016} 9.999902! 8.327114]11.672886 |47| 
7.609853] 9.999996) 7.609857/12.390143]| 8.332924) 9.999899} 8.333025]11.666975 |46) 
7.639816} 9.999996) 7 639820/12.: 8.338753| 9.999897} 8.338856/11.661144/45 
7.667845} 9.999995) 7.667849 8.344504). 9.999894) 8.344610}11.655990/44! 
7.694173} 9.999995] 7.694179] 12.505821|| 8.350181] 9.999891] 8.350289/11.64971 1/43] 
7.718997] 9.999994) 7.719003 97|| 8.355783] 9.999888) 8.355895|11.644105|42) 


7.742478) 9.999993] 77494841 12.257516]| 8 361315] 9.999885] 8.3614.30111.638570)41| 
9.999993] 7.764761|12.235239|| 8.366777] $.999882| 8.366895111.633105|40 
9.999992) 7.785951) 12,214049]| 8.372171] 9.999879) 8.372299|11.627708/39| 
9.999991} 7.806155|t2.193545|| 8.377499) 9.999876) 8.377629]11.622378/38) 
9.999990) 7,825460)12,174540]| 8.382762]. 9.999873] 8.3828399|11.617111|37 
9.999989| 7.845944 8.387962| 9.999870! 8.388092/11.611908/36 


9.999989] 7.861674] 12.138326]] 8.393101} 9.999867] 8.393234/11.606766/35) 
9.999988} 7.870708 8.398179] 9.999864) 8.398315|11.601685134/ 
9.999987) 7.895099]12:104901]| 8.403199] 9.999861) 8.403338]1 1.596662133 
9.999986! 7.910894|12.089106|| 3.408:61| 9.999858) 8.408304)11.591696132) 
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9.999982! 7.955100 8.422717} 9-999848! 8.422869|11.577231129 
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CON AUKwWDmO! 


7.982233 
7.995198 
8.007787 
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8.031919 
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| Cosine. | Sine. | Cotan. | Tang. | Cosine. |~ Sine. 


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89 Deg. ¥ 88 Deg. aed 








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55 
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| 8.636776 


8.645428) 9.999575 


8.673080} 9.999518 


Cosine. 


5 7 = — I NID I DES SILI TE SIO . = ae tea ieteatiieimsiinnnn cxteeteesteenerste enna ee 7 
ae 2 Deg. ; 


Cosine. 


8.542819| 9 999735 
8.546422! 9.999731 
8.549995} 9.999726 
8.553539} 9.999722 
8.557054) 9.999717 


Sine. 








8.560540) 9.999713} 


8.563999) 9.999708 


8.567431| 9.999704 
8.570836 2.09600] 
8.574214! 

8.577566 
8.580822 
8.584193 


887409 
8.590721 
8.593948 
8.597152) 
8.600332 
8.603489 


8.606623 
8.609734, 
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9,999694 
9.999689| 
9.099685 
9.999680 


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9.999670! 
9.999665 
9.999660 
9.999655 
9,999650 


9.999645 
9,999640 
9.999695 
9, 999629] 
9. 999624) 
9.999619 


9.999614 
9.999608 
9.999603 
9 999597 
9.999592 
8.659680! 9.999586 


8.642563) 9.999581 

















8.615891 
8.618937 
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8.648274) 9.999570 
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8.653911} 9.999558 
8.656702) 9.999553 


8.659475! 9.999547 
8.662230) 9.999541 
8.664968; 9.999535 
8.667689, 9.999529 
8.670393| 9.999524 

















8.675751) 9999512: 
8.678405! 9.999506 
8.681043} 9.999500 
8.683665, 9.999493 
8.686272) 9.999487 
8.688863! 9.999481 


8.691438} 9.999475 
8.693998} 9.999469 





8.701589) 9.999450 
8.704090; 9.999443 


8.706577| 9.999437 
8.709049} 9.999431 
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8.713952! 9.999418 
8.716383! 9.999411 
8.718800] 9. 999404 


~ Sine. 








87 Deg. 


8.543084 
8.550268 


8.567727, 


‘8 670870) 11.329130 


LOG. SINES, TANGENTS, &. 






















3 Deg. 
Sine. | Cosine. _ Tang. 5._| Cotang. 


8.718800) 9.999404 871939615. 13.280604)60 | 
8.721204) 9.999398} 8.721806] 11.278194159 | 
8.723595| 9.999501! 8. 724204111,275796158 | 
8.725979| 9.999384| 8.726588] 11.273412157 | 
8.728337] 9. 999978) 8.728959|11.271041 (56 | 
8.730688| 9.999371, 8.731317/11,268683/55| 
8.7330271 9,999364| 8.733663 11,266337154 


8.735354 9.999357! 8.735996) 1126400453] 
8.737667| 9.999350] 8.738317|11,261683152 
8.739969} 0.999343 $ 740626|11. spb aeh | 
8, 742264) 9 9,999336| 8.742922) 11.257078|50] 
8. 744599 9.999329} 8.745207 !11, sea : 


| Cotang. 


11.45691 TT 
1.453309 
11°449732 
8.553817|11.446183 
8.557316! 11.442664, 
oe Sl 


Tang. 





8.546691 











8.564291} 11.435709 


11.432273 
11:428865 
11.425480 | | 


8.571137 
8.574520 
8.577877 
8.581208 
8.584514 











|11.422123 
11.418792 
11.415486 


3.587795) 11.41.2205 
8.59105! iiaoert7 





8. 746802| 9.999322] 8.747479|11.252521|48 | 


8, 749055| 9.999315! 8. 74974011 ! eee 
8.751297| 9.999308| 8.751989/11.248011 146 | 
8.753528| 9.999301} 8.754297|11.2 ae 














8.594.283) 11-4057 17 
8.597492|11.402508 
8 600677| 1.399523 | 








8.757955) 9.999287! 8 758668) a 24133214 
8.760151} 9.999379) 8 .760872| 11.239128)4 


8.762337] 9.9999721 8.769065 |11.23693514 
8,764511| 9.999265| 8.765246|11.234754 10) 
8. 166675) 9.999257! 8.767417|11.232583/39 | 
| 8. 768828 9999250) 8 769578) 11.230422/38 
8.770970| 9.999242| 8.71797 (11228973137 
8.773101| 9.999235| 8.773866|11.226134(36 | 


8.775223] 9.999227| 8.775995|11.224005|35 | 
8.777.333] 9.999220] 8.778114|11.221 886/34 
8.779434) 9 999219] 8.780229] 11.219778/33 | 
| 8. ras 9.999205 : 8.782320! 11,217680'32| 


8. arts A sa038 8. Medea 11,243547|4 


8.603839! 11.396161 


§.606978}11.493022 
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8.6 hia 357018 
8.645853/11.354147) | 
8 648704) 11.351296 
8.651537! 11.348463 
8:654352| 11.345648 
8.657149] 11.342851 


8.659928) 11.340072)) 
8.662689|11.337311 
8.665433) 11.334567 
8.668160! 11.331840 








8.783605| 9.999197| 8.784408|11.215592|31 | 
8.785675| 9.999189] 8.786486) 11.213514|30 | 


8.787736| 9.999181 | 8.788554|11.211446|29 
8.780787| 9.999174] 8.79061 3|11.200337|28 
8.791828] 9.999166] 8.792662'11.207338)27 | 
8.793859! 9 999158| 8.794701|11.205299/26 | 
8.795881| 9.999150). 8.796731 |11.203269|25 | 
8.797894) 9.999142) 8.798752) 11.201248)24 


8.799897| 9.999134 8.800763]11.199237/23 | 
8.801892} 9.999126) 8.802765|11.197235 - 
8.803876) 9.999118) 8.804758] 11.195242)21 | 
8.805852! 9.999110! 8806742111 190888 05 
8.807819| 9.999102| 8 808717|11.191283)191 
8.809777| 9,999094| 8.810683) t1.189517|18} 


8.811726| 9.999086] 8.812641 |11.187359|17 | 
8.813667) 9.999077 8.814589) 11. stats 














8.673563) 11. 326437 


8.676239 /11,323761 
8.678900) 11.321160 











8.681544'11.318456|; 8 815599] 9.999069} 8.816529)11.183471;15 | 
8.684172/11.315828|| 8.817522| 9.999061 | 8.818461/11.181539114| 
8.686784/11.313216}; 8.819436) 9.999053] 8.820384] 11. 179616|13; 
8.689391/11.310619 | 8.821343] 9. 999044) | 8: nite manga tp 


8.691965} 11.308037); 8.823240] 9. 999036, 8.824205 |41 wonky: 
8.694529) 11.305471]} 8.825130} 9.999027) 8. Lapin hel iach 
11.302919}} 8.827011! 9. 999019 8.827992/11. 172008) 
11.300383}| 8. 828884 9.999010) 8.829874|11. 170126| § 
11.297861|| 8.830749] 9.999002/ 8.831748/11.168252| 7 
11,295354.| 8.832607) 9.998993) 8.833613)11.166387; 6 


1.292860]! 8.834436] 9.998984) 8.835471)11. 64520) 5 5 
11.290382!| 8.836207! 9.998976, 8.837321 '11-162679 
dl 88381301 9.998967| 8.830163! 1.160837 ; 

















8.702139 
8.704646 


8.767140 
8.709618 
8.712083 

















8.714534) 11,285466|| 8.839956] 9.998958) 8.840998] 1 1.159002| ¢ 

8.841774} 9.998950) 8.842825|11.157175] 1. 

8,843585| 9.998041) 8.844644) 11.155536 0 

Cosine. Sine. | Cotan. - ~ Tang. 
86 Deg. 








8.716972! 11.283028 
8.719396|11. a 


“Cotan, | Tange. 

















- ars 





LOG. SINES, TANGENTS, &c. 


eect ate erence cae estan ttm NIE ct aA A ET A I LE NS OE LS EDEL EE 


4 Deg. 5 Deg. . 
Sine. Cosine. | Tang. | Cotang. Sine. | Cosine. | ‘Tang. | Cotang. 


| 0} 8.843585) 9.998041) 8 844644)11.155356)) 8.940296) 9.098544) 8.041959|11.0980%8)60 
| 1! 8.845387] 9.998932 sbi 6511-15954 8.941738! 9.998333| 8.943404 1.056596) 
| ol 8.871831 9.999929| 8.848260111.1517401| 8.943174| 99983221 8.944859111.055148158 
| 3| 8.848971| 9.998914| 8.850057|41.149943|| 8.944606| 9.998311| 8.946295|11.053705|57 
| 4) 8.850751) 9.998905) 8.851816/11-148154|| 8.946034) 9.998500] 8.947754)1 1.052266 56 
| 5! 8.852525] 9,998896| 8.853628] 11.146372)| 8.947456! 9.998289) 8.949168) 11.050832155 | 
| 6| 8.854901{ 89988871 8.855403111.1445971| 8.948874) 9.998977! 8.950597|11.049403154 


| 7! 8.856049} 9.998878) 8.857171/11.142829|| 8.950287! 9.998266). 8.952021 | 11.047979/53 
| s| 8'857801| 9.998869| 8.858932|11.141068|| 8.951696] 9.998255| 8.953441|11.046559|52 
| 9| 8.859546| 9.998860| 8.860686|11.139314|| 8.953100| 9.998243] 8.954856|11.045144|51 
| 10) 8.861283) 9.998851) 8,862433/11.137567)| 8.954409) 9.998252| 8.956267|11.045733|50 
11! 8.863014} 9.998841) 8.864173/11.135827), 8 955894) 9.998220) 8.957674) 11.042326/49 
| 12| 8.864738| 9.9988521 8.865906|11.134094!| 8.957284] 9.998209) 8.959075|11.040925148 


'13| s,866455| 9.998803! $.867632111.1323681| 8.958670! 9 998197! 8.960373111.039597/47 
}14| $.868165| 9.998813| 8.869351|11.130649|| 8.960052] 9.998186] 8.961866|11.038134|46 
|15| 8.869868 9.998804) 8.871064|11.128936|| 8.961429) 9.998174| 8,965255|11.036745]45 
}16) 8.871565| 9.998795) 8.872770/11.127250) 8.962801) 9.998165) 8.964659) 11035361 j 4 
17| 3.873255! 9.998785! §.874469111.125531!! 8.964170| 9.998151! $.966019|11.033981 143 
: 8.874938|.9,998776| 8.876162|11.123838|| 5,965534| 9.998139] 8.967594] 1.032606/42 


19| 8.876615) 9.998766 8.877849|11.1221511| 8.946893! 9,9981281 8.968766|11.031834/41 
120} 8.878285| 9.998757| 8.879529|11.120471|| 8.968249] 9.998116] 8,970133|11.029867|40 
21| 8.879949] 9.998747| 8,881202/11.118798)| 8.969600] 9.098104) 8.971496)11.028504|39 
122) 8.881607) 9.908758) 8.882869,11.117131)) 8.070947) 9.998092) 8.979855 11.027145)38 
|'23| $.889258| 9.998728| 8.884530|11.1154701| 8.972289! 9.998080! $.974200!11.025791 187 
|24| 8.884903] 9.998718] 8.886185/11.113815|| 8.973628| shee: 8,975560|11.024440|36 


95| 8.886542 9.998708] 8.887833|11.1121671! 8.974962] 9.980561 8.976906|11.093094135 
126| 8.888174| 9.998699| 8.889476) 11.110524|| 8.976293] 9.998044| 8.978248|11.021752|54 
8.889801} 9.998689) 8.891112/11.108888)| 8.977619) 9.998052} 8.979586)11.020414/33 
ss9tdal| agus0r9 8,899742/11.107258) 8.978941) 9.998020) 8.980021 j11,019079 59 
31 

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8.893035|- 9.998669] 8,894.366|11.105634|! 8.980259} 9.998008! $,982951111.017749 
130| 8.894643] 9.998659] 8.895984)11.104016]| 8.981573] 9.997996] 8.983577111.016423|50 


31| 8.896246| 9.998649| 8,897596]1 1.102404| 8.982883] 9,997984| 8.984899|11.015101 129 
}32} 8.897842| 9.998639) 8.899203|11.100797|| 8.984189] 9.997972| 8.986217|11,013783|28 
| 33 san. 9.998629/ 8 900803/11.099197}| 8.985491) 9.997059) 8.987532|11.012468 27 









34) 8.901017) 9.998619} 8.902398!11.0976021| 8.986789] 9.997947) 8.988842!11.011158) 26 
35] 8.902596] 9.998609| 8.903987|11.096013|| 8.988083] 9.997955| 8.9901 49|11.009851|25 
}36| 8.904169] 9.998599} 8.905570)11,094430|| 8,989374| 9.997922) 8.991451|11.008549|24 


37| 8.905736| 9.998589| 8.907147|11.092855|| 8.990660| 9.997910] 8.999750|11.007250]23 } 
438] 8.907297| 9.998578] 8.908719)11.091281|| 8991943] 9.997897] 8.994045|11,005955|22 
39} 8.908853) 9.998568) 8.910285 11.0897 15)) 8.993222) 9.997885) 8.995397) 11.004663/21 | 
40] 8.910404) 9.999558| 8.911846|11.0881541! g.094497| 9.9978721 s:996604111.003876120 
}a1| 8.911949] 9.998548| 8.913401|11.086599|| 8.995768| 9.997860] 8.997908|11.002092|19 
42| 8.913488] 9.998537] 8.91495 1)11.085049 | 8,997036| 9.997847| 8,9991$8|11.000812/18 


43| 8.915022] 9.998527| 8.916495]11.083505|| 8.998299| 9.997835] 9.000465|10.999535|17 
|44| 8.916550} 9.998516) 8.918034/11.081966)| 8.999560] 9.997822 
45 8.018075 9.998506) $.919568)11,080452)| 9.000816) 9.097809) 9.003007) 10.096995)15 
}46| 8.919591| 9.998495] 8.921096|11.0789041| 9.002069 9.997971 9.004972110.995728|14 
47| 8.929103] 9.998485| 8.922619/11.077381|| 9.003318| 9.997784] 9.005534| 10,994466|13 


48} 8.922610) 9.998474 8.924136 11.075864. | sedate 9.997771) 9.006792 10993208) 12 


49} 8.924112] 9.998464 8.925649 11.07.4951 | 9,005805| 9.997758] 9.008047|10,991953|11 
50) 8.925609) 9.998453) 8.927156)11.072844)| 9.007044 9.997745) 9.009298|10.990702|10 
}51| 8.927100) 9.998442! §-928658/11.071342), 9.008278} 9.997732} 9.010546)10,989454) 9 
52) 8.928587| 9.998451) 8.930155|11.069845|| 9.009510] 9.997719| 9.011790|10.988210| 8 
53] 8.930068] 9.998421| 8.931647111.068353]| 9.010737} 9.997706) 9.013031|10.986969| 7 
}54) 8.931544) 9.998410) 8.933154) 11.066866)| 9.011962) 9.997693) ere hat 6 





























9.001 738|10.998262 16 

















}55| 8.933015] 9.998399} 8.054516)11.065384|| 9.013182) 9.997680] 9,015502|10.084498| 5. 
56} 8.994481) 9.998588) 8.936093)11.065907)) 9.014400, 9.997667) 9.016732)10.985268) 4 
_}57| 8.935942| 919989771 8.937565|11.062435|| 9 0156131 9.997684) 9,017959110.982041| 3| 
158| 8.937398| 9.998366] 8.939092|11.060968|| 9.016824| 9.997641| 9.019183|10.980817| 2. 
159] 8.938850] 9.998355) 8.940494} 11.059506!| 9.018031] 9.997628! 9.020403|10.979597] 1 
160} 8.940296} 9.998544] 8,941952)11.058048]| 9.019235 fae eget 0 





ee ee fee 


Fy Cosine. Sine. ! Gotan. | Tang. Cosine. | Sine. Cotan. | Tang. | | 
85 Deg. ” 84 Deg. . 








LOG. SINES, TANGENTS. &c. 





6 Deg. 





7 Deg. 





Sine, | Cosine Tang. | Cotang. | 


0| 9.019235) 9.997614) 9.021620)10.978380 
1} 9.020435} 9,997601} 9.022834\10,977166 
al 9.021632| 9.997588| 9.024044 10,975956, 
3| 9.022825} 9.997574 9.025251|10.974749) 
4| 9.024016) 9.997561] 9,026455}10,973545 


5| 9.025203 eee | peaks op pepe 
nace 





9.026386] 9.997534! 9.028852110,971148 


7) 9.097567! 9.997520} 9.030046110.969954 
$| 9,028744| 9.997607| 9.031237|10.968763 
9| 9.029918} 9.997493} 9,032425| 

10} 9.031089) 9,997480| 9,039609]10.966591 
ul 9.032957! 9.997466! 9.034791'10.965209 
12 9.033421| 9.974521 9.035969|10.964031 


13! 9.034582! 0.997439! 9.0371 44/10:9629856 
14| 9.035741| 9.997425| 9.038316] 10.961684 
15 9.036896, 9.997411 9.0394.85|10.960515 
9.038048) 9.997397) 9.040651 10.959349 


10.967575 


17| 9.039197! 9.997383} 9.041813!10.958187 
181 9.040342] 9.997369 9,042973|10.957027 


19! 9.041485! 9.997355 9.044130] 10.955870 
20| 9.042625| 9.997341| 9.045284|10,954716 
/21| 9.043762| 9.997527] 9.046434) 10.953566 
22} 9.044895| 9.997318, 9.047582)10,952418 
23\ 9.046026 9.497299} 9.048797 |10.951273 
24| 9.047154| 9.997285| 9.049869|10.950131 


251 9.048277! 9.997271| 9.051008|10,948992 
26| 9.049400| 9.997257| 9.052144|10.947856 
27\ 9.050519) 9.997242] 9.053277] 10.946723 
28! 9.051635} 9.997228! 9.054407|10,945593| 
29 9.052749! 9.997214| 9.055535|10,944465 
30| 9.053859| 9.997199] 9.056659|10.943341 


31} 9.054966) 9.997185| 9.057781 
32| 9.056071| 9.997170] 9.058900|10.941100, 
33) 9.037172) 9.997156) 9.060016 |10.939984 
34} 9.058271! 9.997141! 9.061130!10.938870 
35| 9.059367! 9.997197 
36| 9.060460] 9.997112] 9.063348) 10.936652 


(37| 9.081551| 9.997098| 9.064453! 1 0.935547] 
{38 9.062639 9.997083 | 9,065556 | 10934444 
39} 9.065724). 9:997068) 9.066655)10.933545 


140 9.064806 9,997053| 9.067752!10,93224.8 





se ne a RE I IT LE ITE DES LEELA, 
SS. BE > — 














ne ge ES ES SE ES 





,}42| 9.066962) 9.997024| 9.069938] 10.930062 


|43| 9.068036} 9.997099| 9.071027|10,998973 


44) 9.069107) 9.996994) 9.072113 10,027887 
145} 9.070176! 9.996979] 9.073197) 10.926803 

46| 9.071249| 9.996964] 9.074278! 

47| 9.072506| 9.996949| 9.075356 10,924644 

48| 9.075366] 9.996934) 9.076452/10.923563 


49| 9,074424| 9,996919| 9.077505|10.922495 
50| 9.075480) 9.996904) 9.078576) 10.921424 
51) 9.076533) 9.996889) 9.079644) 10.920556 
52| 9.077583) 9.996874| 9.080710|10.919290 
53| 9.078631| 9.996858} 9.081773] 10,91 8227 
. 154) 9.079676) 9.996843) 9.082853)10.917167 


55 9.080719] 9.996828 9.083891 /10.916109 
56 doscriy 9.996812) 9.084947 10814000 








57| 9.082797| 9.996797! 9.086000|10.914000 
58| 9.085832) 9.996782| 9.087050)10.912950 
459! 9.084864! 9.996766! 9.088098'10.911902! 
60|-9.085894 9.996751| 9.089144 


Cosine: | Sine. | Cotan.— 
» $3. Deg. 














10.942219) | 


9.062240] 10,937760 | 


eu 9.065885| 9,997039| 9.068846! ;0.9311541 


10925722) 


i) SNe. ‘Cosine. 





9.085894 9.996751 
9.086922! 9.996735 
9.087947| 9.996720 
9.088970| 9.996764 
9,089990) 9.996688 
9.091006) 9.996673! 
9.092024 9.996657 

| 
9.093037! 9.996641 | 
9,094047| 9.996625, 
9.095056| 9.996610. 
9.096062) 9.996594! 
9.097065 9.96578 
9,098066| 9.996562 


9.099065! 9.996546 
9.100062] 9.996530) 
9.101056] 9.996514 
9.102048 9.996498 
9,103037; 9.996482 
9.104025] 9.996465 


9,105010| 9.996449 
9.105992] 9.996433 
9.106973] 9.996417 
9.197951); 9.996400. 
9,108927| 9.996384. 
9.109901} 9.996368, 





9,110873| 9.996351] 


9,111842| 9.996335 
9.112809} 9.996318 
9.113774} 9.996302 
9.114737| 9.996285 
9,115698| 9.996269 


9.116656] 9.996252 
9.117613} 9.996235 
9.118567) 9.996219 
9.419519! 9.996202 
9.120469] 9.996185 
9.121417| 9.996168 


9.122362} 9.996151 
9.123306] 9.996134 
9.124248) 9.996117 
9.125187} 9,996100 
9,126125| 9.996083 
9,127068| 9.996066 


9.127993] 9.996049 
9,128925| 9.996032 
9.129854) 9.9960" 5 
9.130781] 9.995998 
9.131706| 9.995980 
9.182630) 9.995963 


9.133551| 9.995946 
9.154470) 9.995928 
9.135387) 9.995911 
9.136303) 9.995894 
9.137216| 9.995876 
9.138128) 9.995859 


9.139037| 9.995841 
9.139944) 9.995823 
9.140850) 9.995806 
9.141754} 9.995788 
9.142655) 9.995771 











10.910856}| 9.143555} 9.995753 
Tang. Cosine. | Sine. 
82. Deg. 





Tang. | Cotang. 
9.089144) 10.910856) 60 
9,090187|10.909813/59} 
9,091228) 10.908772158 
9.092262! 10.907734157 |} 
9.093309) 10.90669 8/56 
9.094336}10:905 664/55 | 
9,095367|10.904633|54| 


9.09639 5| 10.903605|5S 
9.097 4.22] 10.902578|52 
9.0984.48|10.901554|5% 
9.099468} 10.900532|50 
9,106487|10.899513|49 
9,101504/10.898496)48 


9,102519|10.897481|47 
9,103532| 10896468146 
9.104549] 10895458145 
9.105550! 10, 8944.50) 44, 
9.106556|10.893444143 
9.107559] 10,892441}42 


9,108560\10.891440141 
9,109559|10.890441) 40 
9,110556|10.88944.4] 39 
9.111551|10.888449|38 
9,11254.3|10.887457137 
9,113533|10.886467/36 


9.114521) 10.885479)35 
9.115507|10.884493/34 
9.116491 |10.883509|33) 
9.117472|10.882528)32 
9.118452) 10.881548/31 
9,119429)10.880571 


9,1204.04) 10,879596/29 
9.121377] 10.878623}28 | 
9.122348] 10.877652) 27 
.9,123317|10.876683/26 
9,124.984110.875716)25 | 
9.125249 10,87 4751 Q4; 


9.12621 1}10.873789123 
9.127172|10.872828|22 
9, 128130) 10.871870124 
9.129087] 10,87091 3120 
9.130041] 10,869959|19 
9.130994) 10.869096|18 


9.131944) 19.868056|17 
9.132893] 10.867107|16 
9.135839|10.866161)15 
9.1347 84110.86521 6114 
9.1357 26| 10.86427 4/13 
9,136667|10.863333 12 


9.137605] 10.862395'11 
9/138542110.861458 10} 
9.439476)10.860524| 9} 

/9,140409110.859591| 8 
9,141340/10,858660_- 
9,14.2269}10.857731 


7 
6 
9.143196) 10,856804 34 
9.144.121)10.855879} 4. 
9.145044) 10854956} 3): 
2 
1 
(8) 










9.145966'10,854054: 
9,146885|10.835115 
9.147 803|10. 852197 


] 
Cotan. “| Tang. | | 


et tr 


LOG: SINES, TANGENTS, &. 






ase 



































































Sine. Cosine. 
9.143555) 9.995753 
9.144453} 9.995735 
9.145349} 9.995717 
9.146243). 9.995699 
9.147136} 9.995681 
9.148026) 9.995664 
9.148915} 9.995646 


9.449802} 9.995628 
9.150686] 9.995610 
9.151569} 9.995591 
9.152451, 9.995578 
9153330} 9.995555 
9.154208) 9.995537 


13) 9.155083} 9.995519 
14! 9.155957) 9.995501 
15| 9.156830) 9.995482 
16! 9.157700! 9,995464 
17| 9.158569! 9.995446 
£8] 9.159435] 9.995497 


19) 9.160301) 9.995409 
20) 9.161164! 9.995390 
ZL! 9,162025| 9.995372 
22) 9.162885] 9.995353 
23! 9.163743) 9.995334 
24! 9.464600) 9.995316 


25} 9.165454 9.995297 
26) 9,166307| 9.995278 
27) 9.167159! 9.995260 
28| 9.168008! 9.995241 
29! 9.168856) 9.995222 
30| 9.469702) 9.995205 


3t) 9.170547] 9.995184 
32) 9.171389) 9.995165 
33) 9.172230) 9.995146 
34| 9.173070} 9.995127 
35) 9.175908) 9.995108 
36) 9.174744) 9.995089 


Bs 
37| 9.175578] 9.995070 
38| 9.176411) 9.995051 
39} 9.177242] 9.995032 
40| 9,178072) 9.995013 
44) 9.178900] 9.994993 
42] 9.179726] 9.994974 


43! 9.180551| 9.994955 
44) 9,181374| 9.994935 
45] 9,182196| 9.994916 
46] 9.183016] 9.994896 
47| 9,183834| 9.994877 
48} 9.18465 11. 9.994857 


Tang. | Cotang. Sine, »| Cosine. 
9.147803) 10.8521971) 9.194332) 9.994620 
9.1487 18] 10.8512821} 9.195129} 9.994600 
9.149632) 10.850368| | 9.195925] 9.994580 
9.150544] 10.849456}| 9.196719] 9.994560 
9.151454) 10.848546]| 9.197511} 9.994540 
9.152363] 10.8476371| 9.198302] 9.994519 
9.153269) 10,8467321}| 9.199091! 9.994499 


9.154174} 10,845826]| 9.199879] 9.994479 
9.155077|10,844923]| 9.200666] 9.994459 
9.155978} £0.844022!| 9.201451} 9.994438 
9 156877|10.843123]! 9.202234! 9.994418 
9 157775|10.842225|| 9:203017] 9.994398 
9.15867 1}10,841329]| 9,203797] 9.994377 


9159565) 10.840435|| 9.204577) 9.994357 | 
9.160457} 10.839543]! 9.205354) 9.994336 
9.161347] 10.838653|| 9.206131) 9.994316 
9.162236) 10,837764)| 9.206906} 9.994295 
9.163123} 10.836877|| 9.207679] 9.994274 
| 9. 164008} 10,.835992)| 9.208452) 9.994254 


9,164892|10.835108}| 9.209232) 9.994233 
9.165774) 10.834226|| 9.209992] 9.994212) 
9.166654) 10.833346|| 9.210760] 9.994191 
9.167532]10.832468]| 9.211526} 9.994171 
9.168409/10.831591)| 9.212291} 9.994150 
9,169284/ 10.830756}| 9.213055] 9.994129 


9.170157|10.829843]| 9.213818) 9.994108 
9.171029) 10.828971]| 9.214579} 9.994087 
9.171899) 10.82810t}| 9.215338] 9.994066 
9.172767 |10.827233|| 9.216097) 9.994045 
9.173634) 10.836366|| 9.216854] 9.994024 
9.174499] 0.825501 || 9.217609] 9.994003 


9.175362) 10.824638]| 9.218363} 9.993982 
9.176224|10.823776|| 9.219116] 9.993960 
9.177084) 10.822916]| 9-219868| 9.993939 
9.177942} 10.822058]| 9.220618} 9.993918 
9.178799) 10.821 :201}| 9.221367) 9.993897 
9,179655|10.820345]| 9.222115] 9.995875 


9.180508] 10.819492)| 9.222861} 9.993854 
9.181360) 10.818640}| 9.223606) 9.993832 
9.182211} 10.817789]| 9.224349) 9.993811 
9.183059] 10.8169411| 9.225092) 9.993789 
9.183907|10.816093}| 9.225833) 9.993768 
9.184752) 10.815248}| 9.226573} 9.993746 


9,185597|10,.814403]} 9.227311} 9.993725 
9.1864.39|10.813561}| 9.228048} 9.993703 
9,187280)10.812720)| 9.228784} 9.993681 
-9,188120}10.811880]) 9.229518) 9.993660} 
9,188958|10.811042}| 9.230252] 9.995638 
9,189794)10,810206]| 9.230984) 9.993616 


9}, 9.185466| 9.994838) 9.190629] 10.809371|| 9.231715] 9.993594 
50} 9.186280} 9.994818] 9.191462] 10.808538|| 9.232444! 9.993572 
. 15} 9.187092] 9.994798] 9.192294] 10.807706|| 9.233172] 9.993550. 

52| 9.187903) 9.994779} 9.193124|10.806876|| 9.233899] 9.993528 
53| 9.188712) 9.994759) 9.193953110.8060471| 9.234625] 9.993506 
54) 9.189519} 9.994739] 9.194780110,805220]| 9.235349] 9.993484, 


55| 9.190325! 9.994720} 9.195606] 10.8043941| 9.236073] 9.993462 
56) 9.191130| 9.994700] 9.196430110.8035701| 9.236795] 9.995440] 9,213454|10.756646 
10.802747|| 9.237515! 9.993418] 9,244097|10.755903 
10.801926!| 9.238235] 9.993396! 9:244839|10.75516! 
10.801106| 9.238953 eset oaise70|t0.75442 | 


Cotang. 


























9.201 345/10.798655| 58 
9.2021 59}10.797841 57 
9.202951) £0.797029| 56} 
9 203782) 10,796218)55 
9.204592) 10.795408)54 


9.205400} 10.794400)53 
9.206207|10.793793|52 
9.207013} 10.792987|51} 
9,207 817 | 40.792183|50 
9.208619) 10.791 38149 
9.209420) 10.790580}48 


9.210220] 10.789780|47 
9.211018] 10.788982)|46 
9,211815|10.788185|45} 
9,212611|10.787389| 44. 
9,213405|10.786595}43 
9.214198] 10.785802|42 


9.214989) 10.785011|4t 

9.2157 80)10.784220|40 
9.216568) 10.783432|39 
9.217356) 10.782644138} 
9.218142] 10.781858|37 
9.218926) 10.781074| 36} 


9.2197 10}10.780290/35 
9.220492! 10.789508| 34 
9,221272|10.778728| 33 
9,222052|10.777948| 32 
9,222830|10.777170| 31 
9.223607|10.776393| 30 


9.224389] 10.775618| 29] 
9.225156|10.7749441 28 
9.225999] 0.77407 1| 27 
9.226700) 10.773300| 26 
9,227471|10.772529| 25} 
9.228239|10.771761 (241 


9.229007) 10.770993|23 
9.229773} 10.770227 |22 


OM Anmopwmol 





=) 


Bes. 







9.23434.5|10.765655 
9.235103) 10.764897 


9,236614|10.763386 
9.237368)|10.762632 


9.238120 10.761880|11 
9.238872! 10.761 128] 10 
9.239622! 10:760378 
9,24037 1 10.759629 
9,241118|10.758882 
9.241865! 10.758135| 


9.242610) 10.757390 





57\ 9.191933} 9.994680} 9.197253 
fs 9.192734 sani oa 


Bik mha ANos. 


591 9.193534 9.994640] 9.198894 
60 os 9.994620} 9.199713 


eT aera. 








ce 


10.800287!) 9.239670! 9.993351] 9,246319110.753681 


Sine. | Cotan. | Tang. | Coun Sine. a Gotan, 1 g- 
84 Deg. 80 Deg: ‘ 








11 Deg. 


Cot GaRwHomol — 





1 9.267395 


LOG, SINES, TANCENTS, &c. 


10 Deg. 


Cosine. 


9.993351 
9.993329 
9.993307 
9.993284 
9.993262 
9.993240 
9.993217 


9.993195 
9.993172 
9.993149 
9.993127 
9.993104 
9.993081 


Sine. 


9.239670 
9.240386 
0.241101 
9.241814 
9.242526 
9.243237 
9.243947 


9.244656 
9.245363 
9.246069 
9.246775 
9.247478 
9.248181 


9.248883 
9.249583 
9.250282: 
9.250980 
9.251677 
9.252373 


9.253067 
9.253761 
9.254453 
9.255144 
9.255834 
9,256523 


9.257211 
9.257898 
9.258583 
9,259268 
9.259951 
9.260633 


9.261314 
9.261994 
9.262673 
9.263351 
9.264027 
9.264.703 


9.265377 
9.26605! 
9,266723 





9.993036 
9.993013 
9.992990 
9.992967 
9.992944 
9.99292) 
9.992898 
9,992875 
9.992852 
9.992829 
9.992806 


9.992785 
9.992759 
9.9927 36 
9,992713 
§ 992690 
9.992666 


9.992643 
9.992619 
9.992596 
9.992572 
9,992549 
9992525 


9,992501 
9.992478 
9.992454 
9.992430 
9.992406 
9.992382 


9.992359 
9.992335 
9.992311 
9.992287 
9.992265 
9.99223G 
9.992214 


9.992190 
9.992166 


9,268065 
9.268734 


9.269402 
9.270069 
9.270735 
9.271400 
9.272064. 
9.272726 


9.273388 
9.274049 
9.274708 
9.275367 
9.276025 
9.276681 


9.077337 
9.977991 
9.278645 
sar 
| 


9.992118 
9.992093 


9,992069 
9.992044 
9.992020 


9,279948) 9.991971 


60 9,280599 9.992941 9.288652 
Cos 





osine, hy Sine. 





sip aa 


9.993059) 


9.262292 


9.992142) 


Tang. Cotang. 


9,246319|10.753681 


9.247057 | 10.752943 
9.247794) 10752206 
9,248530|10.751470 
9,249 264! 10.7507 36 
9,249998|10.750002 
9.250730] 1.749270 


9.251461110.748539 
9.2521 91 |10.747809 
9,252920)10.747080 
9.253648] 10.746352 
9,25437 4.) 10.745626 
9.255100) 10,744800 


9.255824) 10.744176 


9.256547 |10.743453 


9.257269) 10.74.2731 


9.258710) 10.741296 
9,259429|10.74057 1 


9.260146 |10.759854 
9.260863) 10.739137 
9.261578|10.738422 
10.737708 
10.736995 
10.736285 


10.735572 
10,734862 
10.734153 
10.7.33445 
10.7327 39 
10.732033 


10.731329 
10,730625 
10.729923 
10,729221 
§.271479)10.728521 
9272178) 1.727822 


§ 272876) 10,727124 
9,273573|10.726427 
9.274269} 10.725731 
9,274964|10.725036 
9,275658)10.724342 
9.276351/10.723649 


9.277043] 10.722957 


9,265005 
9.263717 


9,264428 
9.265138 
9,265 847 
92266555 
9267261 
9.267967 


9.268671 
9.269375 
9,270077 
9.270779 





9.277734) 10.722266 
'9.9784.24|10.721576 


9.279113] 10,7208 87 
9.279801 | 10.7201 99 
9.280488/10.719512 


9.281174|10.7 18826: 
9.281858/10.718142 
9,282542/10.717458 
9.283225 |10:716775 
9.283907 |10.716093 
9.284588) 10.715412 


9.285268 |10,714732 
9.285947110.714053 
9.286624110.713376 
9,287977(10.712023 
10,711348 


“Tang. 





“Cotan. 








Cosine. 


“9.991947 
9.991922 
9.991897 


Sine. 


9,280599 
9.281.248) 
9,281897 
9.282544) 9,991873 
| 9.283190} 9.991848 
9.283836) 











| 9.991823 
9.284480) 9.991799 


9, 285124, 9.991774 
9. 285766. 9.991749 
9.286408 9 991724 
9.287048 9.991699 
9.287688 9.991674 
9. eenpae 9,991649 


9.288964) 9.991624 
9 "289600 | 9.991599 
9,290236 9.991574 
9.290870, 9.991549 
9.291504 9.991524 
9,292137| 9.991498 


9.292768) 9.991473 
9.293399! 9.991448 
9.294029 9.991492 
9.294658, 9.991397 
9,295286, 
9.295913) 9.991346 


9,296539) 9.991321 
9.297164) 9.991295 
9.297788 
9.298412 
9,299034' 9.991218 
9,299655| 9.991193 


9.300276! 9.991167 
9,300895| 9.991141 
9,501514) 9.999115 
9.302132) 9.991090 
9.302748 9.991064 
9.303364) 9.991038 


9.803979] 9.991012 
9.304593] 9.990986 
9.305207 
9,305819 
9.306430 


9.990934. 
9.990908 





9.307650} 9.990855 
9.308259 
9.308867 
9,309474, 
9.310080! 
9.310685 


9.311289 
9.311893 
9.312495 
9.313997 
9,313698} 9.990591 
9.314297 | 9.990565 


9.314897 | 9.990538 
9,315495! 9.990511 
9.316092} 9.990485} 


9.307041) 9.990882 | 


9.990803 
9.990777 
9.990750 
9.990724 


9.990697 
9.990671 


9.990618 





9,990431 | 
9.990404 


Sine: _ 


9.317284 
| 9.317879 


II “Cosine, 








BE cork MBSA pe i etn 1081 Pea aed Od OE cia SOR tad ieremtil on eran 2b 


79 Deg. 78 Deg. 


aay saves 





9.991372) 9,303914|10.696086|37 


9,991270| 9.306519}10.693481134 
9.991244! 9.307168]10,692832(32 


9.308463 |10.691537 (30 


9.990960! 





9.990829! 








9,990645| 9.321851|10 678149 






























Tang. | Cotang. 7 | 


9.288652|10.711328|60 
9.289326] 10.710674159} 
9.289999] 10,710901158 
9,290671)10.709329|57 
9,291342|10.708658156 
9,299013|10.767987 155 
9.292682) 10.707318 54 


9.293350) 10 706650 )53 
9.294017) 10.705983|52 
9.294684)10,705316)54 
9.295349} 10.704651 150 
9.296013] 10,703987 |49 
9.296677 |19.703323 148 


9.297339|10,702661 |47. 
9.298001 |10.701999 146 
9.298662) 10.701338/45 
9.299322) 10.700678|44 

9.299980) 10.700020/43 
9.300638) 10 699362/42 


9.301295 110.698705 | 42 
9.301951 |10,698049140 
9.302607 |10.697393|39] 
9.303261 | 10.6967 39/38 








9.304567 |10.695433|38 


9.30521 8] 10.694782|35 
9.305869) 10.694131 (34 


9.307 816]10,692184|38 








9.369109) 10.690891 | 
9.309754) 10,690246 128 
9.300399 | 10.689601 |27 
9.311042/10,688958/26 
9.311685/10.688315 125 | 
9.312327 |10.687673/24) 


9.312968] 10,687032/23 
9.313608] 10.686392|22 
9.314247 |10,68575312i 
9.314885 | 10.685115|20 
9.315525|10-684477|19| 
9.3161 59|10.683841118 


9.316795 |10.683205|17 
9.317430} 10.682570/16 
9,31 8064) 10.68 1936)15 
9,.318697/10 681303|14 
9.319330) 10. 680670}18 
9,51996 1) 10.680039|12 


9.32059 :2|10.679408|11: 
9.321 222/10.678778 10 


9.322479] 10.677521) 
9.323106 | 10.676894 
9.323733) £0.676267 


9.324358 |10.675642 
9.324983 )10.675017 
9.325607 |10, 674393) 


Cos & HN Ow ¢ 


9.991996) 9.287301 |10.7126991/ 9.316689} 9.990458] 9.326231 /10. 673769 A 


9 326853 
9, 327475 


“Gotan. 


10.673147 
10.672525 


Tang. _ 


0 


——— 
Hl 














ete 


LOG. SINES, TANGENTS, &c: 





12 12 Deg. 


Sine. 
9.317879 
9 318473 

9.319066 
9.319658 
9320249 
9.320840 
9.821430 


9.322019] 
9.322607 
9.323194 
9.323780 
95324366 
9.324950 


9.325534. 
9.326111 
9.326700 
9.327281 
9.327862 
9.328442 


9.529021 | 
9.329599 
9.330176 | 
9.330753 
9.331329, 
9.331903) 


9.332478 
9.333051 
9.333624 
9.334195 
9.334767 
9.335337 


9.335906 
9.336475 
9.537043 
9.337610 
9.358176 
9.338742 


9.339307 
9.339871 
9.340434 
, 9.340996 
41) 9.341558 
9.342119 


9.342679 
9.343239 
9.343797 
9.344.355 
147| 9.344912 
9.345469 


9.346024 
9.346579 
9.347134 
9.347687 | 
9.348240 
9,348792 


9.349343 
56] 9.549893 
9.350443 
9.350992 
9.351540 

9.352088 


Cosine. 
ee 


9 990404 
9.990378 
9.990351 
9.990324. 
9.990297 
9.990270 
9.990243 


9.990215 
9.990188 
9.990161 
9.990134) 
9,990107| 
9.990079 


9.990052 
9.990025 
9.989997 
9.989970 
9.989942 
9.989915 


9.989887 
9.989860 
9.989852 
9.989804 
9.989777 
9.989749 


9.989721 
9.989693 
9.989665! 


i 
oocwen Aue wrwmol 


ppt 
to tom 


14 


9.989610! 


9.989553 
9.989525 
9.989497 


9.989441 
9.989413 


9.989385 
9.989356 
9.989328 
9.989300 
9.989271 
9.989243 


9.989214. 
9989186 
9.989157 
9.989128 
9.989100 
9.989071 


9.989042 
9,989014 








9.988956 


9,988898 
9.988869 


9.988811 
9.988782 
9.988753 
9.988724 





9.989582! 





9.988985) 


9.988997] 9, 


9.988840) 





‘osine.” | Sine. 











ne. |. Tang. Teeny 
9.327475 | 10.672525 


9.328095 |10.671905] 


9.328715 |10.671285 
9,329334.|10.670666 
9,329953 | 10,670047 
9.330570 | L0.6694.30 
9.331187 | 10.668813 


9.331803 /10.668197 
9.332418} 10.667582 
9.333033 | 10.6669 67 
9.333646 | 10.666354 
9.334259 |10.665741 
9.334871 |10.665129 


9,335482|10.664518 
9.336093] 10.663907 
9.336702] 10.663298 
9.337311 }10.662689 
9.337919] 10.662081 
9.338527 | 10.661473 


9.339133] 10.660867 
9.339739] 10.660261 
9,340344|10.659656 
9.340948! 10.659052 
9.341552] 10.658448 
9 342155 |10.657845 


9 342757 10.657243) | 


9, 343358 | 10.656642 
9 343959) 10.656042 


9. 989637) 9.344558! 10.655442 


9.345157 | 10 654843 
9.345755 | 10.654245 


9.346353) 10.653647 
9.34.6949|10.653051 
9.347545110.652455 


D; 989469) 9,34.8141 |10.651859 


9.348735 |10.651265 
9.349329] 10.650671 


9,.34.9922110 650078 
9.350514}10.649486 
9.351106/10.6 48894. 


9.351697 |10.648303) 


9.352287 |10.647713 
9.352876 |10.647124 


9.353465 | 10.6465 35 











9.354053} 10.64.5947 
9.354640} 10.645360 
9.355227 |10.644773 
9.355813) 10,6441 87 
9.356398 | 10,643602 


9.356982) 10.64.3018 
9.357566) 10,642434. 


9.358149} 10.64.1851) 


'9.358781| 10.64.1269 
359313] 10,640687 
9.359893|10,640107 


9.360474! 10.639526 
53|10,638947 
2|10,638368 





9.3627 8/,10,637213 








9.363364 10.636636 
Cotan. 


“Yang. 


77 Deg. 


Se ar Se Sarees sass 


9.36 
9.361 
9.3622 710 637790) 


Sine. 


“9.352088 
9.352635 
9.353181 
9.353726 
9.35427 I 
9.354815 
9.955358 


9.355901 
9.356443 
9.356984 
9.357524 
9.358064 
9.358603 


9.359141 
9.359678 
9.360215 
9.360752 


"9.361287 


9.361822 


9.362356 
9.362889 
9.363422 
9.363954 
9.364485 
9.365016 


9.365546 
9.366075 
9.366604 
9.367131 
9.367659 
9.368185 


9.368711 


9.369236 
9.369761 
9.370285 
9.370808 
9.371330 


9,371852\ 
9.372373 
9.372894 
9.373414 
9.373933 
9.374452 


9.374970 
9.375487 
9.376003 
9.376519 
9.377035 
9.377549 


9.378063 


9.378577 
9.379089 
9.579601 
9.380118 
9.380624. 





9.381134 


9.381643 
9,382152 
9.382661 
9.383168 
9.383675 


Cosine. 








13. Deg. 
_ Tang. _ 
9 988724 


Cosine. 


9.988695 
9.988666 
9.988636 


9.988607 
9.988578} 


9.988548 


9.988519 
9.988489 
9.988460 
9.988430 
9.988401 
9.988371 


9.988342 
9.988312 
9.988282 
9.988252 
9,988223 
9.988193 


9.988163 
9.988133 
9.988103 
9.988073 
9.988043 
9.988013 


9.987983 
9.987953 
9.987922 
9 987892 
9.987862 
9.987832 


9.987801 
9.987771 
9.987740 
9.987710 
9.987679 
9.987649 


9.987618 
9.987583 
9.987557 
9.987526 
9.987496 
9.987465 


9.987434 
9.987408 
9.987372 
9.987341 

9.987310 
9.9872791 


9.987248 
9.987217 
9.987186 
9.987 155 


9.987124 


9.987092 


9.987061 
9.987030 
9.986998 
9.986967 
9.986936 
9.986904 


_ Sine. 


9.395694 10.604306 


Cotan. 


Cotang. 


9.363364, 10.636636 x 
® 363940/10.636060|59 
9.364515/10,635485|58 
9,365090|10.634910|57 
9.365664|10, 634336) 5 6 
9.366237 |10.633763) 55 
9.366810)10,633190|54 


9.367382) 10.632618|53 
9.367953|10.632047 |52 
9.368524/10.631476|51 
9.369094) 10.630906] 50 
9.369663|10.630337 |49 
9.370232) 10,629768)48 


9.370799 |10.629201 |47 
9.371367 |10.628633|46 
9.37 1933/10. 628067 |45: 
9.372499}10.627501 | 44. 
9.373064) 10.626936| 43 
9.373629)10.626371 |42 


9.374193}10.625807 |41 
9.3747 56|10.625244)40 
9.375319)|10.62468 1 |39: 
9.375881 |10.624119|38 
9.376442) 10.62355 8/37 
9.377003}10.622997/36 


9.377563) 10.622437|35 
9.3781 22| 10.621 878134, 
9.378681 |10.62:319133 
9,379239}1 0.620761 |32 
9 379797 |10.620203}31 
9.380354) 10.619646}30 


9.380910]10.619090)29 
9.381466|10.618534| 28 
9,382020}10.617980)27 
9.382575 |10.617425|26 
9.383129|10.616871)25 
9,383682/10.616318)24 


9.384234 10.615766|23 
9.3847 86/10.615214 29! 
9.385337 |10.614663} 2 
9.385888 10.614112)20 
9.386438)10.61 3562/19 


-9,586987110.613013|18 


9.387536/10.612464, 17 
9.388084|10.611916/ 16 
9.388651|10.611369) 15 
9.339178]10.610822|14 
9.389124)10.610276|13 
9.39027 0|10.619730|12 


9,390815|10.609185)11 
9.391 360) 10.608640]10 
9.391903) 10.608097| 9 
9.3924.47|10.607553| 8 
9.392989/10.607011| 7 
9,393531|10.606469 


9.394073} 10.605927 
9.394614, 10.605586 
9.395 15410.604846). 


10.603767 
10.603229 | 


Waa. ¥ 


9.396233 
9.396771 





- 76 Deg 


LOG, SINES, ‘TANGENTS, &c. 








12 
13! 
14 
15 
16 
17 
18| 
19 









22 





25 
26 








28 
29 
30 

31 
32 








34, 











56 
57 
58| 





24 





Sine. 





9.383675| 


9.384182 
9.384687 
9.385192 


9.385697 9.986778; 


9.386201 


9.386704! 9.986714 
9.387207 | 


9.387709 
9.588210 
9.3887 11 
9.389211 
9.389711 


9.390210 
9.390708 
9.391206 
9.391703 
9.392199 
9.392695 


9-394179 


9.394.673 


31 9.395166 
9.395658 


9.396150 
9.396641 


27| 9.397132 


9.397621 
9.398111 
9.398600 


9.399088 
9.599575 


3) 9.400062 








9.986531 


9.986299 
9.986266 
9.986234 


anato4| 





| 9.986137 
9.985942 


9.985909 
9.985876 


14 Deg. 


Cosine. 


9.986904 
9.986873 


9,986841 
9.986809 


9.986746 


9.986683 
9.986651 
9.986619 
9.986587 
9.986555 
9.986523 


9.986491 
9.986459 
9.986427 
9.986395 
9.986363 


9.986202 
9.986169 


9.986104. 
9.986072 
9.986039 


e 986007 


9,9§5974 


9.985843 


9.400549} 9.985811 


9.401035 
9.401520 


9.402005 
9.402489 
9.402972 
9.403455 
9.403938 
9,4.04.420 


9.404901 
9.405382 
9.405862 
9.406341 


9.406820) 


9.407299 


9.407777 
9.408254 
9,408731 
9,409207 
9.409682 


| 9.410157 
5| 9.410632 


9.411106 
9.411579 
9.412052 





Cosine.” 


~ Vou. il. 














‘9.985480 

















9.985778 
9.985745 


9.985712 
9.985679 
9.985646 
9.985613 


9.985580 


9.985547 
9.985514 


9.985447 
9.985414 
9.985381 
9.985347 


9.985314 
9.985280 
9.985947 
9.985213 
9.985180 
9.985146 


9,985 13 
9.985079 
9.983045 
9.985011 


59! 9.412524] 9.984978 
60 9.412996) 9.984944 


Sine. | 



































Tang. 

9.396771 
9.397309 
9.397846 
9.598383 
9.398919 
9.399455 





9.400524 
9.401058 
9401591 
9.402124 
9402656 
9,403187 


9.403718 
9.404249 
9.404778 
9.405308 
9.405836 
9,406364 


9.406892 
9.407419 
9.407945 
9.40847 1 
9.408996 
9.409521 


9.410045 
9.410569 
9.411092 
9.411615 
9.412137 
9.412658 


9.413179 
9.413699 
9.414219 
9.414738 
9.415257 
9.415775) 


9.416293 
9.416810 
9.417326 
9.417842 
9.418358 
9.418873 


9.419387 
9.419901 
9.420415 
9.420927 
9.421440 


7| 9.421952 


9.422463 
9,4.22074 
9.493484 
9.423993 
9.424503 
9425011 


9.425519 
9.426027 
9.426534. 
9.427041}. 
9:4.97547 
9. 428052 


| “Gotan. 








75 Deg. # 


10.603229) 


9.399990) 


10,583190 


-10,581127 


10,578560 


10.574481 
10,575973 





Cotang. | 





10,602691 | 
10.6021 54 
10.601617 
10.601081 

10.600545 
10.6000L0 


10.599476 
10.598942 
10.598409 
10.597876 
10.597344. 
10,596813 


10.596282 
10.595751) 
10.595229 
10.594692 
10.5941 64 
10,593636 


£0,593108 
10.592581 
10.592055 
10,591529 
10,591004 
10,590479 


10.589955| 
10.589431 
10.588908 
10,588385 
10,587863 
10,587342 


10,586821 
10,586301 
10,585781) 
10585262 
10,584743 
10,584.225 


10,583707 








10.582674 
10.582158 
10.581642 


10,580613 
10-580099 
10,579585| 
10,579073) 





10,578048 


10,577537 
10.577026 
10.576516 
10.576007 
10.575497! 
10.574989 








10,573469 
10.572959|' 
10.57.2453! 
10. 571948) 


(Pang. 7 4 Ce 








D 





Sine, 

9.412996 
9.413467 
9.413938 
9.414.408 
9.414878 
9,415347 
9.415815 


9.416283 
9.416751 
9.417217 
9.417684 
9.418150 
9.41 8615 


9,419079 
419544 
9,420007 
9.420470 
9.420933]. 
9.421395 


9.421857 
9.422318 
9.422778 
9.423238 
9,4.235697 
9.424156 


9.424615 
9.425073 
9.425530 
9.425987 
9.426443 
9,426899 


9.497354 
9.427809 
9.428263 


9.428717 
“9,429170 


9.429623 


9.430075} 
9.430597 


9.430978 


9.431429 
9.431879 
9,432329 


9.432778 
9.433226 
9.433675 
9,4.34122 
9.434569 


9, is7ai9l 
9.437686 


9,4.38129 
9.438572 
9.439014 
9.459456 
9.439897 


' “Cosine.” 





15 Deg. 


Cosine. 


9.984944 
9.984910 
9.984876 
9 984849 
9.984808 
9,984774 
9.984740 


9.984706 
9.984672 
9.984638 
9.984603 
9.984569 
9.984536 


9.984500 
9.984466 
9.984432 
9.984397 
9.984.363 
9.984328 


9.984294. 
9.984259 


9.984924} 


9.984190 
9.984155 
9.984126 


9.984085 
9.984050 
9.984015 
9.983981 


9.983946] 


9.983911 


9.983875 
9.983840 
9.983805 
9.983770 
9.983735 
9.983700 


9.983664 
9.983629 
9.983594 
9.983558 
9.983523 
9.983487 


9.983452 
9.983416 
9.983381 
9.983545 
9,983309 
9.985273 


9.983238 
9.983202 
9.983166 
9.983130 
9.983094 
9.983058 


9.983022 
9.982986 
9.982950 
9.982014. 
9.982878 


| 9.440338} 9.982842 
Sine. 







































9.439563 


9.445923 


9.453187 


Tang. , Cotang. | 

"9.428052!" 10.571948)/60 
9.428558} 10.571442|59 
9.429062} 10.570938(58 
9.429566! 10.570434|57 
9.430070} 10.569930|56. 
9.430573] 10.569427|55 
9.431075] 10.568925)\54 
9.431577| 10.568423/53 
9,432070| 10.567921/52 
9,432580| 10.567420 51 





9.433080} 10.566920)50 
0,433580} 10.546426)\49 
9.434080 10,565920/48 


9.434579) 10,.5654.21|47 
9,4.35078) 10.564922/46 
9.435576] 10.564424)45 
9.436073) 10.£63927 |44: 
9.456570] 10,563430|43 
9,437067| 10.562933/42 


9.437563} 10.562437|41 
9.458059] 10.561 94140 
9.438554) 10.56144.6/39} 
9.439048] 10.560952|38 
10.560457/37 
10.559964/36 


10.559471135 
10.558978|34 
10.558486|33 
19.557994)32 
10,557503)51 
10,557012130 


10.556521/29 
10,556032)28 
10,555542197 
10.555053|26 
10554565) 25 
10,554077/24 


10.55.35 89/23 
10;553102)/22 
10.552616/21 
10.552190)20 
‘10,55164419 
(10.551159)18 


10,55067 4) 17 
10,550190]16 
10549706] 15 
10.549223/14 
10,548740|18 
10,548257|12 


10.547775)11 
10.547294| 10 
10.546813 
10,546332 
10.545852 
10,545372 
10.54.4893} : 
10.544414 
10.54.3936 
10.543458 
10.542981 
10.542504 

Tang. 




















9.440036 


9.440529 
9.441022 
9.441514 
9.442006 
9.442497 
9.442988 


9.443479 
9.443968 
9,4444.58 
9.444947 
9.445435 
























9.44641! 
9.446898 
9 4.47384 
9.447870 
9.448356 
9.448341 


9.449326 
9.449810. 
9.450294. 
9.450777} 
9.451260 
9.451743 


9.452225 
9,452706 


9.453668 
9.454148 
9.454.628 


9.455107 
9.455586 
9,456064 
9.456542 
9.457019 
9.457496 





Cotan. 


Oo a nena: 


a 


OoOonN Aamwtd 


Sine. 


9.440338 
9.440778 
9.441218 
9.441658 
9.442096 
9.442535 
9.442973 


9.443410 
9.443847 
9.444984 
9.4447 20 
9.445155 
9.445590 


9.446025 
9.446459 
9.446893 
9.447326 
9.447759 
9.448191 


9 448623 
9.449054. 


9.449485 
| 9.449915 


9.450345 
9.450775 


9.451204 
9.451632 
9.452060 
9.452488 
9.452915 
9.453342 


9.453768 
9.454194 


9.455044, 
9.455469 


pete 


9.455893 


9.456316 
9.456739 
9.457162 
9.457584 
9,458006 
9.458427 


9.458848 
9.459268 
9.459688 


. 9.460108 


9.460527 
9.460946 


9.461364 
9.461782 
9.462199 
9.462616 
9.463032 
9.463448 


5| 9.463864 


9.464279 
9.464694 
‘9.465108 
9.465522 


9.465935 
1 Cosine. | 











LOG. SINES, TANGENTS, &c. 


16 Deg. 
Tang. 


Cosine. Se 
9,982842| 9.457496: 
0,982805; 9.457975 
9.982769) 9.458449 
9.982733! 9.458925 
9.982696! 9.459400 
253660 9.459875 


9.982624! 9.460349 


| Cotang. 
10.54.2504 


0.540195 


eee 


9.982587 
9.982551 
9.982514 
9.982477 
9.982441 
9.982404 
9.982367 
9.982331 
9.982294 
9.982257 
9.982220 
9.982183. 


9.982146 
9.982109 
9.982072 
9.982035 
9981998 
9.981961 


9.981924 
9.981886 
9.981849 
9.981812 
9.981774 
9.981737 


9.981700 
| 9.981662 
9.981626 
9.981587 
9.981549 
gosta 








9.981474 
9.981436 
9.981509 
9.981361 
9.981323 
9.981285 


an 








9.981209 
9.981171 
9.981133 
9.981095 
9.981057 


9.981019 
9.980981 
9.980942 
9.980904 
9.980866 
9.980827 


9.980789 
9.980750 
9.980712 
9.980673} 
9.980635 
9.980596 


Sine. 

















9.460823 
9.461297 
9.461770 
9.462242 
9.462715 
9.463186 


9.463658 
9.464128 
9.4.6 4599 
9.465069 
9.465539 
9.466008 


9.466477 
9466945 
9.467413 
9.467880) 
9.468347 
9.468814 
9.469280 
9.469746 
9.470211 

9.470676 
9.471141 

9.471605 


9.47 2069 
9.472532 
9.472995 
9.473457 
9.473919 
9.474381 


9.474842 
9.475303 
9.475763 
9,476223 
9.476683 
9,477 142 


9.477601 
9.478059 
9.478517 
9.478975 
9.479432 
9.479889 


9.480345 
9.480801 
9.481257 
9.481712 
9.482167 
9.482621 


9.483075 
9.483529 
9.485982 
9.484435 
9,484,887 
9.485359 


Cotan. 


73 Deg. 


'10.530720 


Tang. 





10.54.2027 
10,541551 
10.54)075 
10.54.0600}; 


10.539651 


10.539177 
10.538703 
10.538230. 
10.537758 
10.537285; 
10:536824 


10,536342 
10.595872 
0.535401 
10,534931 
10,534461 
10:533992 


10.533523 
10.533055 
10.532587 
10.5321 20 
10.531653 
10.531186 


10.530254 
10.5297 89 
10.529324 
10.528859 
10.528395 


10.527931 
10.527468 
10.527005 
10.526543 
10.526081 
10,525619 


10.525158 
10.5 24697 
10.5 24.237 
10.528777 
10.523317 
10.522858 


10.522399 
10.521944 
10.521483 
10.521025 
10.520568 
10.520111 


10.519655 | 
10,519199 
10.518743 
10.518288 
10,517833 
10.517379 


10.516925). 
10.516471 

10.516018; 
Salve 





0.515113! 


10,514661 





Sine: 


17 Deg. 


———$—$ 


9.465935 


' 9.466348 


9.466761 
9.467173 
9.467585 
9.467996 
9.468407 


9.468817 
9.469297 
9.469637 
9.470046 
9.470456 
9.470863 


9.471271 
9.471679 
9.472086 
9.472492 
9.472898 
9.473304 


9.473710 
9.474115 
9.474519 
9.474923 
9.475327 
9.475730 


9.476133 
9.476536 
9.476938 
9.477340 
9.477741 
9.478142 


9.478542 
9.478942 
9.479342 
9 479741 
9.480140 
9.480539 


9.480937 
9.481334 
9.481731 
9.482128 
9.482525 
9.482921 


9.483316 
9.483712 
9.484107 
9.484501 


9.484895) 


9.485289 


9.485682 
9.486075 
9.486467 
9.486860 
9.487251 
9.487643 


9.488034 
9.488424 
9.488814 
9.489204. 
9.489593 
9.489982 


Cosine. 





A yp ES SS 





Cosine: 

9.980596 
9.980558 
9.980519 
9.980480 
9,98044.2 
9.980403 
9.980364 


9.980325 
9.980286 
9.980247 
9.980208 
9.980169 
9.980130 


9.980691 
9.980052 
9.980012 
9.979973 
9.979934 
9.979895 


9.979855 
9.979816 
9.979776 
9.97975 

9.979697 
9.979658 


9.979618 
9.979579 
9.979539 
9.979499 
9.979459 
9.979420 


9 979380 
9.97 9340 
9.979300 
9.979260 
9.979220 
9.979180 


9.979140 
9.979100 
9.979059 
9.979019 
9.978979 
9.978939 


9.978898 
9.978858 
9.978817 
9.978777 
9.978737 
9.978696 


9,97 8655 
9.978615 
9.978574 
9.978533 
9.978493 
9.978452 


9.978411 
9.978370 
9.978329 
9.978288 
9.978247 
9.978206 


Sine. 





U2 Deg. 








Tang. _ Cotang. 





9.485339|10.514661 \60 


9.485791 | 10.514209}, 
9.486242) 10.513758 
9.486693} 10.5 13307 
9.487143} 10.512857 


9.487593] 10.512407 [55 


9.488043) £0.511957 


9.488492) 10.511508 
9.488941 |10.511059 
9.489390) 10.510610 
9,489838/10.510162 
9.490286 | 10.5097 14, 
9.490733) 10.509267 


9.491180] 10.505820 
9.491627] 10.508373 
9.492073]10.507927 
9,492519110,507481 
9.492965 | 10.507035 
9,493410/10.506590 


9.493854) 10.506146 
9.494299} 10.505701 
9.494743] 10,505257 
9.495186] 10,504814 
9.495630] 10.504370 
9.496073] 10.503927 


9.496515] 10.503485 
9.496957] 10.505043 
9.497399] 10.502601 
9.497841] 10,502159 
9 498289] 10.501718 
9.498722) 10.501278 


9.499163) 10.500837 
9.499603) 10.500397 
9.500042] 10,499958 
9.5004.81 | 10.499519 
500920} 10,499080 
9.501359} 10.498641 


- 9.501797} 10.498203 
9.502235|10,497765 
9.502672! 10.497328 
9.503109} 10.496891 
9.503546] 10.496454 
9.503982) 10,496018 


9.504418] 10,495582 
9.504854) 10,495146 
9.505289) 10,494711 
9.5057 24|10,494276| 
9.506159] 10,49 3844 
9.506593} 10.493407 


9.507027 | 10.492973 
9.507460} 10.492540 
9.507893) 10.492107 
9.508326} 10.491674 
9.508759} 10,491241 
9.509191) 10.491.809 


9:509622| 10.490378 


9.510054] 10.489946} 


9.510485] 10489515 
9.51091 6}10.489084 
9.511346|10.488654 
9.511776] 10,488224 


Cotan. | Tang. 




















59 
58 
57 
56 


54 


53 
52 
51 
50), 
49 

48 
47 
46 
45 
44, 
4 
42 


iM 
40 
39 
38 
37}. 
36 
35] 
34 
33) 
32|- 
3L 
301 


29 
28} 
o7| 
26 
25 
24 


93 
22 
21}. 
20: 
19 
18), 
17 
16) 
15, 
14) 
12 


11 


i 


| 


[Ss 


= : 
Coomn Anewwomol e 


—— am 
Be GO ko 


ee 
Oo eNan 


{29 








| 9.499204 


9.504485 


Cosine. 


LOG. SINES, TANGENTS, &c. 


18 Deg. 


A 


Sine. Cosine. 


9.489982} 9.978206 
9.490371} 9.978165 
9.490759) 9.978124 
9.491147) 9.978083 
9.491535} 9.978042 
9.491922) 9.978001 
9.492308} 9.977959 


9.492695} 9.977918 
9.495081) 9.977877 
9.493466} 9.977835 
9.493851) 9.977794 
9.494236} 9.977752 
9.494621; 9.977711 


9.495005] 9.977669 
9.495388) 9.977628 
9.495772) 9.977586 
9,496154| 9.977544 
9.496537| 9.977503 
9.496919) 9.977461 


9.497301! 9.977419 
9.497682) $.977377 
9.498064; 9.977335 
9.498444) 9.977293 
9.498825) 9.977251 
9.977209 


9.977167 
9.977125 
9.977083 
9.977041 


9.499584 
9.499963 
9.500342 
9.500721 
9.501099} 9.976999 
9.501476| 9.976957 


9.501854) 9.976914 
9.502231} 9.976872 
9.502607] 9.976830 
9.502984) 9.976787 
9.503360} 9.976745 
9.503735| 9.976702 


9.504110} 9.976660 
9.976617 
9.504860} 9.976574 
9.505234) 9.976532 
9.505608| 9.976489 
9.505981| 9.976446 


9.506354) 9.976404 
9.506727) 9.976361 
9.507099) 9.976318 
9.507471) 9.976275 
9 507843} 9.976232 
9.508214} 9.976189 


9.508585 | 9.976146, 
9.508956 
9.509326) 9.976060. 
9.509696} 9.976017; 
9.510065| 9.975974 
9.510434} 9.975930 


9.510803} 9.975887 
9.511172} 9.975844 
9.511540} 9.975800 
9.511967} 9.975757 








9,5 12275| 9.975714! 9.536561 
9.536972 


9.512642) 9.975670 





9.976103 9.5 


Cotang. 


Tang... 
9.511776 | 
9.512206) 10.487794 
9.512635 |10.487365 
9.513064 10.486936 
9.513493|10.486507 
9.513921 |10.486079 
astapaa)foAss651 


9.514777|10.485223 
9.515204) 10.4847 96 
9.515681 | 10.484369 
9.516057 10.483943 
9.516484! 10.483516 
9.516910|10.483090 


9.517335|10,482665 
9.517761 |10.482239 
9,518186|/10.481814 
9.518610) 10.481390 
9.519034. 10.480966 
9.519458|10,480542 


9,519982'10.480118 
9.520305 |10.479695 
9.520728) 10.479272 
9.521151) 10.47 884.9 
9521573} 10,478427 
9.521995|10.478005 


9.522417} 10.477583 
9.522838] 10.477162 
9,523259| 10476741 
9523680} 1 0.476320 
9.524100110.475900 
9.524520) 10,475480 


9.524940! 10.475060 
9,525 360|10.474641 
9.525778] 10.474222 
9,526197|10.473803 
9.526615] 10:473385 
9.527033|10.472967 


9.597451| 10472549 
9.527868] 10.472132 
9.528285)10.471715 
9.528702} 10,47 1298 
9,529119110.470881 
9.529535] 10.470465 


9.52995 1|10.470049 
9.530366! 10.469634 
9.53078 1|10.469219 
9.531196] 10.468804 
9.531611 |10.468389 
9,632025 | 10,467975| 


9.532439 |10.467561 

32853110.4671 47 
9,533266|10.466754 
9.533679|10.466321 
9.534092) 10.465908 
9,534504110.465496 


9.53.49 16|10,465084, 
9 535328|10,464672 











9.535739)\10.464261)| 9.533009 


10.463850 
10,463439 
10.463028 


a sdeeke 





Sine. 
10.488224|| 9.512642 


9.513009 
9.513375 
9.513741 
9.514107 
9.514472 
9.514837 


9.515202 
9.515566 
9.515930 
9.516294 
9,516657 
9.517020 


9.517382 
9.517745 
9.518107 
9.518468 
9.518829 
9.519190 


9.519551 
9.519911 
9.520271 
9.520631 
9.520990 
9.521349 


9.521707 
9.522066 
9.522424) 
9.522781 | 
9,523138 


9.523495 


9.523852 
9.524.208 
9.524564 
9, 524920 
9,525275 
9.525630 


9.525984 
9.526339 
9.526693 
9.527046 
9.527400 
9.527753 


9.528 105 
9.528458 
9.528810 
A\.529161 
9 529513 
9.529864. 


9 530215 
9.530565 
9.530915 
9.531265 
9.531614 
9.531963 


9.532312 
9.532661 





9.533357 
9.533704 


19 Deg. 


Cosine. 


9.975670 


9.975627 
9.975583 
9.975539 
9.975496 
9.975452 
9.975408 


§.975365 
9.975321 
9.975277 
9.975233 
9.975189 
9.975140 


9.975101 
9.975057 
9.975013 
9.974969 
9.974925 
9.974.880 


9.974836 
9.974792 
9.974748 
9.974703 
9.974659 
9.974614 


9.974570 


9.974525 
9.974481 
9.974436 
9 974391 
9.974347 


9.974302 
9.974257 
9.974212 
9.974167 
9.974122 
9.974077 


9.974032 
9.973987 
‘9 973942 
9.973897 
9.973852 
9.973807 


9.973761 


* 9.973716 


9.973671 
9.973625 
9.975580 
9.973535 
9.973489 
9.973444. 
9.973398 
9.973352 
9.973307 
9.973261 


9.973215 
9.975169 
9.973424 
9.973078 
9.973032 








Tang. Cotang. 


9.537382) 10,462618 1/59 


9.536972! 10,463028 | 


9.537792] 10.462208 158 
9.538202) 10.461798157 | 
9.538611 | 10,461389|56) 
9.539020) 10.460980155) 


9.539429] 10.460571|54 


'9.539837| 10.460163153) 
9.540245] 10.459755 152 
9.540653) 10.459347 151 
9.541061 | 10.458939]50 
9.541468) 10.458532|49 
9,541875| 10.458125|48 


9.542281) 10.457719|47 
9°542688|10.457312146 
9.54.3094| 1 0.456906/45 
9.543499] 10.456501 |44 
9.543905] 10.456095|43 
9,544310|10.455690142 


9.544715 |10.455285| 41 


9.545119)10.454881 


9.545524) 10.45447 6139 
9.545928! 10.454072138 
9.546331 | 10.453669/37; 


9.546735 | 10.453265 


9.547138] 10.452862| 35) 
9.547540] 10.452460/34 
9.547943) 10,450057 {33 


9.548345] 10.461655 
9.548747 | 10,451253 


9,553149| 10.446851 
9.553548] 10,.446452 


9,553946| 10.446054| 18 


9.554344 10.445656| 17 
9,554741 | 10.44.5259) 16 
9.555139) 10.444861/ 15 
9.555536) 10.44.4464) 14 
9,555933|10.444.067|13 
9.556329 | 10443671 |12 


9.556725 | 10443975 |14 
9.557121110,442879|10 


9.557517 |10.442483 
9.557913|10.442087 
9.558308] 10.441692 
9.558703) 10,441297 


9.559097 |10.440903 
9.559491 10.440509 
9,559885|10.430115 
9.560279] 10439721 
9.560673! 10.439327 





9.561066|10.438934| . 


Cotan._ _ Tang. 





























40: 


36! 


82 
31| 


Vo mis Rt Ontos 


i i ry a 


LOG. SINES, TANGENTS, &. 





- 20 Deg. 
Cosine. _ 


9.972986 
9.972940. 
9.972894 
9.972848 
9.972802 
9.972755 
9.972709 


9.972663 
9.972617 
9.972570 
9.972524 
9.972478 
9.972431 


9.972385 
9.972338 
9.972291 
9.972245 
9.972198 
9.972151 


9.972105 
9.972058 
9.972011 
9.971964 
9.971917 
9,971870 


9.971823 
9.971876 
9.971729 
9.971682 
9.971635 
9.971588 


9.971540 
9.971493 
9.97 14.46 
9.971398 


Sine. 
92534052 
9,534745 
9.535092 
9.535438 
9.535783 
9.536129 


9.536474 
9.536818 
9.537163 
9.537507 
9.537851 
9.538194 


5} 9.538538 
9.538880 
9.539225 
9.539565 
9.539907 
9540249 


9.540590 
9.540931 
9.54272 
9.541613 
9.541953 
9.542295 


9.542632 
9.542971 
9.548310 
9.543649 
9.543987 
9.544325 


9.544.663 
9.545000 
9.545538 
9.545674 
9.546011|.9.971351 
9.5 46347) 9.971303 


9.546683! 9.971256 
9.547019) 9.971208 
Q 547354! 9.971161 
9.547689| 9.971113 
9.548024) 9.971066 
9.548359) 9.971018 


9.548693] 9.970970 
9.549027| 9,970922 
9.549360}. 9.970874 
9.549693) 9.970827 
9,550026| 9.970779 
9,550359| 9.970731 


9.550692 9.970685 
9.651024) 9.970635 
9.551356) 9.970586 
9.551687). 9.970538 
9.552018| 9.970490 
9.552349) 9.970442) 


9.552680 9.970394 
9.553010 9.970345 
9.553341} 9.970297; 
| 9.553670) 9.970249 
59] 9.554000} 9.970200 
60 9.554329} 9.970152 9.584177, 10415823, 


“Cosine. | Sine. | Cotans  T ang. Fal 
Pt 


Tang. Cotang 


9.561066) 10,438934 
9.561459} 10.4385 41 
9,561851|10.438149 
9,56224.4| 10.4377 56 
9.562635 |10.437364 
9,563028| 10436972 
9,563419| 10,4365 81 


9.569811] 10436189 
9,564202| 10.435798 
9,564593| 10,435407 
9.564983] 10.435017 
9,565373|10.434627 
9.565763] 10.434237 


9.566153} 10,433847 
9.566542) 10.4.33458 
9.566932] (0.433068 
9.567320! 10,432680 
9.567709] 10432291 | 
9.568098! 10,431 902) 


9,568486|10,431514 
9,568875|10.43 1127 
9.569261] 10. 130739) 
9,569648 | 10,430352) 
‘9.570035 |10.429965 
9.570422! 10.429578 


9.570809}10,429191 
9.571195] 10,4.28805 
9.571581) 10,428419, 
9.571967] 10.428033 
9.572352] 10,427648 
9.572738) 10,427262 


9.573123) 10.426877 
9.573507 |10.4.264.93 
9.573892|10.426108 
Da CaACOHLO Aas 724 
9.574660! 10.4.25340 
9, 575044 10, 424956 


9, S754 27110. 494573 
9.575810) 10,424190 
9.576193} {0.423807 
9.57657 6] 10,493424 
9.576959) 10.4.23044 
9.577341 10.422659 
9.577723) 10,4.22277 
9.578104) 10,421896 
9.578486 LO4215$ 4) 
9.578867 110.421133 
9.579248) 10,420752 
9.579629110,4.20371 


9.580009 10,419991 
9.580389) 10.419611 
9.580769} 10, 419231 
9.581149110.41 8851) 
9.581528| 10.41 8472. 
9.581907110.41 8093 


9.582286] 10.4177 14, 
9,582665|10.417335; 
9.583044) 10.416956 
9.58342: alto, 446578. 
9,583800! 10.41.6200 


oon Anke orol me 


a es 






































a 








9.560531 
9.560855} 





Sine. 
9.554329 
9.554658 
9.554987 
9.555315 
9.555 G43 
9.55597 
9.556299 


9.556626 
9.556953 
9.557280 
9.557606 
9.557932 
9.558258 


9.558583 
9.558909 
9.559234 
9.559558 
9.559883 
9.560207 


9.561178 
9.561501 
9.561824 
9.562146 


9.562468 


9.562790 
9.563112 
9.563433 
9.563755 
9.564075 


9.564396 
9.564716 
9.565936 
9.565356 
9.565676 
9.565995 
9.5663) 4. 
9.566632 
9.566951 
9.567269 
9.567587 
9. 567904 


9, 568222 
9.568539 
9.508856 
9.569172 
9.569488 
9.569804. 


9.570120 
9.570435) 
9.570751 
9.571066) 
9.57 1380 
9.571695 


9,57 2009 
9.57 2323 
9.572636 
9.572950 
9.573263 
91573575 


Cosine. 


9.970152 asgiift |ibatae 415823 6 






9.967217 
9.967166! 


Sine. 





PR ek 9) MOE ROR «<r 


21 Deg. 


Cosine. Pane 





| Corang ae 


9.970105] 9.584555 |10.415445)5 
9,970055| 9,584932110. 415068188 
9.970006 9/885309|]0.414691(67 
9.969957 9.585686 /10.414314|56), 
9.969909] 9,586062/10.413938)55 
9,969860| 9.586439|10.413561|54 


9,0608111 9.686815} 10.41 3185 |53' 
9.969762! 9.587190| 10412810152 
9,969714| 9,587566|10.412434|51 
9.969665| 9.587941 | 10.412059|50 
99969616) 9.588316/10.411684)49} 
9, '969567| 9,588691110.411309/48] | 


9,969518), 9.589066}10,410934)47 
9.969469! 9.589440)10,410560/46 
9.969420} 9,589814|10 410186/45 
9,969370| 9.590188} 10. toamnass i 
9.969921) 9.590562) 10,409438 4 

9.969272 


9.969223! 9.591308!10, ves Al 
9.969173} 9.591681|10,408319/40 
9.969124] 9.592034/10. 407946) 39 
9.969075) 9. pets ty 407574138 
9.969025! 9.592799!10 407201 | 37 
9.968976! 9.595171|10.406829|36 


9.968926! 9, 593542110, 406458135 
9.968877| 9.593914|10.406086/34| 
9.968827| 9.594285140.405715 33 
9.968777| 9.594656140,40534413 

9.968728! 9.595097 | Heneeal, 
9,968678| 9.595398] 10,404602 5 


9,968628| 9,595768|10.404232129 
9,968578| 9.596138| 10,403862|28} 
9.968528] 9.596508|10.4.03492)27 
9,968479} 9.596878! 10,403122)26 
9.968429] 9.597247|10,402753125 
9,968379| 9.597616] 19 402384|24 


9.968329| 9.597985|10:409015|23 
9.968278| 9.598354 10,401646|22 
9.968228) .9.598722/10, 40127821 
9.968178) 9.599091 110,400909!20 

9.968128} 9.599459|10,400541(19 
9.968078] 9.599827 faut 


9,968027} 9,600194| 10,399806|17 
9.967977| 9,600562)10.399438| 16] 
9.967927 9.600929) 10.399071 15 
9.967876! 9.601296!10,39870414 
9.9678:26| 9.601663|10.398337|13 
9.967775} 9.602029 |10,397971]19 


9.967725| 9.602395|10.397605|11 
9.967674) 9.602761 |10.597239)10 
9.967624) 9.603127 )10.396873! 9 
9.967573 9,603493110,396507| 8 
9.967522}. 9,603858|10.396142| 7 
9.967471| 9.604223|10.395777| 6 
5 
4 
3 











































9.967421] 9. 604588 |10.395412 
9.967570) 9.604953) 10.595047 
9.967319 

9.967268 


9,605317'10.394683 

9. 60568210. 394318 2 

9.60046! 10.393954 

9,606410|10.393590 
“Gotan. ib Tang. ma 

68 Deg. 








eae 








Sine. 
9.573575. 
9.573888 
9.574200 
9.574512 

9,574824 
9.575136 
9.575447 


9.575758 
9.576069 
9.576379 
9.576689 
9.576999 
| 9.577309 


9.577618 
9.577927 
9.578236 
9.578545 
9.578853 
9.579162 


9.579470 
9 579777 
9.580085 

- 9.580392 
9.580699 

24) 9.581005 


9.581312 
9,581618 
9.581924. 
9.582229 
9.582535 
9.582840 


9.583145 
9.583449 
9.583754 
9.584058 
9.58436 | 
9.584665 


9.584.968 
9.585272 
9.585574 
9.585877 
9.586179 
9.586482 


| 9.586783 
9.587085 
9.587386 
9.587688 
9.587989 

9.588289 | 


9.588590 
9.588890 
9.589190 
9.589489 
9.589789 
9.590088 


- 9 590387 
9.590686 
9.590984 
9.591282 
9.591580 
9.591878 


Con anewrrmol > 








87} 
58 | 
59 
60 





- Cosine. 








LOG. SINES, TANGENTS, &c. 


22 Deg. 
Tang. | Cotang. | 
166) 9.606410) 10.303590 
9.967115] 9.606773) 10.393227 
9.967064! 9.607137/10.392863 
9.967013| 9.607500] 10 392500 
ety 9.607863) 10.3921 37 


Cosine. 
9.967166 











9.966910) 9.608225  10,391775 
9.966859! 9,608588]10,391412 


9.966808) 9.608950)10.39 1050 
9.966756| 9.609312|10,390688 
9,966705| 9.609674|10.390326 
9.966655) 9.610036) 0.389964 
9.966602} 9.610397 | 10.389603 
9.966550) 9.610759|10.389241 


9 966499! 9.611120/10.388880 
9.966447| 9.611480|10.388520 
9.966395| 9.611841] 10.388159 
9.966344 9.612201 |19.38799 
9.966292! 9.612561/10.3874.39 
9.966240| 9.612921 |10.387079 


9.966188! 9.613281/10.386719 
9.966136| 9.613641 |£0.386359 
9.966085| 9.614000] 10.385000 
9.966035) 9.614359 10.389641 
9.965981! 9.614718/10.385282 
9.965929] 9,615077|10.384923 


9,965876| 9.615435 |10.384565 
9.965824 9.615793] 10.384207 
9,965772) 9.616151 |10.383849 
9.965720] 9.616509|10.383491 
9.965668] 9.616867|10.583138 











-9.965615| 9.617224|10.382776 


9.965563] 9.617582|10.33241 § 
9,965511| 9.617939}10.382061 
9.965458) 9.618295) 10.381705 
9.965406} 9,618652)10.381348 
9,9653531 9.619008] 10,380992 
9.965301] 9.619364| 10.380636 


9.963248| 9-619720 10.380280 
9.965195| 9,620076/10.379924 
9.965143) 9,620452|10.379568 
9.965090! 9.620787110 379213 
9.965037| 9.621142!10.378858 
9.964984| 9.621497|10.378503 


9,964951| 9,621 852|10.378148 
9.964879] 9.622207110.377793 
9.964826) 9.622561/10.377439 
9.964773! 9.622915110.377085 
9,964720| 9.623269|10.376731 


9.964666) 9.623623) 10.376377 


9.964613| 9.623976|10.376024 
9.964560 9.624330] 10,375670 
9.964507} 9.624683/10.375317 
9.964454 9.625036) 10.374964 
9.964400] 9.625388) 10.374612 
9.964547) 9.62574.1/10,374359 


9.964294| 9.626093|10.373907 
9.964240] 9.626445 |10.373555 
9.964187! 9.626797|10.373203 
9.964133! 9.697149110.372851 
9.964080! 9.627501|10 372490| 
: pes 9.627852) 10.372148 


Sine. 























Cotan. | Tang.) | 
CM 5 al Nagel al a ees A 


67 Deg. 


9,608461| 9.960899 





; 23 Deg. : 
Cosine. | Tang. 
9,964026| 9.627852 
9.592176} 9.963972) 9.628203 
9.592473! 9.963919! 9.628554 
9.592770| 9.963865! 9.698905 
9.593067 9.963811! 9.629255 
9.906963) 9.963757 9.629606 
9,593659! 9,963704| 9.629956 


9.593955! 9.963650! 9.630306 
9.59425 11 9.963596| 9.630656 
9,594547| 9.963542| 9.631005 
9.594842) 9.963438] 9.631355 
9,595137| 9.963434! 9.631704 
9.595432| 9.963379} 9.652053 


9.595727! 9,963325| -9,632402 
9.596021| 9.96327 t| 9.632750 
2008 9.963217| 9.633099 































Sine. 


ee Cotang: 
9.591878 


40.372148) 60} 
10.371797| 56 
10,371446] 68 
10.37 1095/57 
10. 370745156 
10.3703941 55 
10,37004415 4 


1036969453 
10:369344.|52 
10.568995151 
10.368644150 
10,368296 14.9 
10,367947 48 


10.367598 |47 
10.367250|46 
10.366901 |45 
10.366553 |44 
10.366205 |43 
10.365857 |42 


10.365510/41 
10.365162|40 
10.364.815/39 
10.364468] 38 
10,364121| 37 
10,363774) 36 


10.363428| 35 
10.353081 |34 
10.362735/33 
10.362389|32 
10,362044/31 
10.361698|30 


10,361355|29 
10.361008|28 
10.360663|27 
10, 36031 8/26 
10, $59973/25 
10.359629|24 


10.359284 
10,358940 
10,358596 
10.358253) 
10.357909 
10,357566) 1) 
10.357223|17 
10.356880} 16 
10.356537| 15 
10.356194) 14 
10.355859) 15 
10.35.5510) 12 


10.355168) 11 
10.354826| 10 
10.354.484. 
10.354143 
10.353801 
10.353460 
10.353119 
10.3527 78| 
10.352438| 3 
10.352097| 2 
10.351757! 1 
10.351417| 0 











9.596609} 9.963163} 9.633447 
9.596903! 9.963108! 9.633795 
9.597196] 9.963054| 9.634143 


9.597490! 9.962999 
9.597783| 9.962945 
9,598075| 9.962890 
9.598368) 9.962836 
9.598660! 9.969781 
9.598952| 9.962797 


9,5992441 9.962672 
9,599536| 9.962617' 
9.599827) 9.962562 
9,600118| 9.962508 
9.600409! 9.962453 
9.606700| 9.962398| 


9,600990| 9.962343 
9.601280] 9.962288 
9,601570| 9.962233 
9.601860} 9.962178} 9.639682 
9.602150) 9.962123) 9.640027 
9.6021439| 9.962067] 9.640371 


9,602728| 9.962012| 9.640716 
9.603017] 9.961957| 9.641060 
9.603805) 9.961902! 9.641404 
9.603594! 9.961846! 9.641747 
9,603882| 9.961791] 9.642091 
9.604170} 9.961735| 9.642434 


9,604457| 9.961680] 9.642777 
9.604745] 9.961624) 9.643120 
9.605032} 9.961 509) 9.643463 
9,605319! 9.961513! 9.643806 
9.605606} 9.961458] 9/6441 48 
9.605892! 9.961402! 9.644490 


9.606179} 9.961346| 9.644832 
9.606465) 9.961290) 9.645174 
9.606751) 9.961235} 9.645516 
9.607036) 9.961179! 9.645357 
9.607322) 9.961123| 9.616199 
9.607607 9.961067) 9.646540 


9.607892; 9.961011] 9.646881 
9,608177| 9,960955) 9.647222 

9.647562 
9.960843| 9.647903 
9.960786] 9.648243 
9.960730| 9.648583 


9.634490 
9.634838 
9.635185 
9.635532 
9.635879 
9.636226 


9.636572 
9.636919 
9.637265 
9.637611 
9.637956 
9.638302 


9.638647 
9,638992 
9.639337 




















AN oro 


eS) 


9.6087 45 | 
9.609029} 
9.609313 





t 











~ Cotan. 
66 Deg. 


Cosine. Sine. 


‘Tang. 





Nye 


: Sine. 

“O! 9.609313 
| 4] 9.609597 
5 9.609880 





- 3} 9.610164 
' 9.610447 
| 9.610729 

9.611012 


9.611294 
9.611576 
, 9} 9.611858 
110} 9.612140 
fii; 9.612421 
9.612702 
i 9.619983 
9.613264 
| 2613sis 
9.613825 
| ootdoss 


| 8 


9.614385 


0 
{ 
12 
I 
14 
5 
16 
Pty. 
TLS 
19) 9.614665 
ot 9.614944 


121! 9.615223 

2) 9.615502 
123| 9.615781 
124; 9.616060 


(25| 9.616338 
26; 9.616616 
a7 9 616894 
128) 9.617172 
» 9.617450 


1 
fe 
2 





| 30) 9.617727 


431) 9.618004 
132] 9.618281 
133] 9.618558 
}34| 9.618834 
$35] 9.619110 
(36, 9.619386 


| 37 
38 
139 
40 
At 
142 


lag 





9.619938 
9.620213 
9.620488 
9.620763 
9.621038 


| 9.621313 
144) 9.621587 
45| 9.621861 
146) 9.622135 











143) 9.622682 


149} 9.622956 
150! 9.623229 
151| 9.623502 
52) 9.623774 
153} 9.624047 
54| 9.624319 


55! 9.624591 
156} 9.624863 
57| 9.625135 
158! 9.625406 
59} 9.625677 
160} 9.625948 








———= 


Cosine. | 


9.619662| 


A7| 9.622409} 


Cosine. 
9.960730 


9,960674| 9.648923 
9.960618} 9,649990/10.350737 


9.960561 
9.960505 
9.960448 
9.960392 


9.960335 
9.960279 
9.960222 
9.960165 
9.960109 
9.960052 


9.959995 
9.959938 
9.959882 
9.959825 
9.959768 
9.959711 


9.959654 
9.959596 
9.959539 
9.959489 
9.959425 
9.959368 


9.959310 
9.959253 
9.959195 
9.959138 
9.959080 
9 959023 


9.958965 
9.958908 
9.958850 
9.958792 
9.958734 
9.958677 


9.958619 
9.958561 
9.958503 
9.958445 
9.958387 
9.958329 


9.958271 
9.958213 
9.958154 
9.958096 
9.958038 
9.957979 


9.957921 
9.957863 
9.957804 
9.957746 
9,957687 
9,957628 


9,957570} 9.667021) 


9.957511 


9.957452) 9-667682 


9.957393 
| 9.957335 
9.957976 


Sine. 





9.656692) 10.343308 


9.662043 


Cotan. 


65 Deg. 
eile wths= ON Pea Rate eco LOR RTE BO NE TL Vi AR 


LOG. SINES; TANGENTS,- &c. 


Cota] 


1.351077 


Tang 
9.648583 


es Le 


9.625948 
i 9.626219 





10,351077 
9.626490 
-9.626760 
9.627036 
| 9.627300 


9.649602, 10.350398| 
9,649942| 10.350058| | 
9.650281| 10.349719 
9,650620) 10.349380 


9.650959, 10.349041 | 


9.627570 


9.627840; 
9.628109 
9.628378 
9.628647 





9.651297|10.348703 
9.651636] 10.348364 
9.651974| 10.348026|| 
9,652312110.347688'| 9.628916 
9,682850|10.3473501| 9.629185 


9.652988 £0.3470121 9.629453 
9.653326 '10,3466741| 9 629721 
9,653663)10.346837|! 9.629989 
9.654000| 10.346000)| 9-630257 
9.654.337 |10.345663/| 9.6305 24 
9.654674) 10.345326!| 9.630792 


9.655011/|10.344989|| 9.631059 
9,655948) 10.544052 2.91535 
9,655684/10.344316|! 9.631593 
10.545980) 
| Lions | 








9.681859 
9.632125 
9.632392 


9.652658 


9.656020 
9.656356 





9.657028! 10.342972 
9 657364| 10.342636|| 9.632923 
9.657699|10.342301'| 9.633189 
9.658034) 10.341966|| 9.633454 
9.658369|10,341681|| 9.633719 
9.658704) 10.34126|} 9.633984 


9,659939]10.3409611| 9.634249 
9.659373|10.340397|| 9.634514 
9,659708/ 10.3402921| 9.634778 
9,660042|10.339958!| 9.635042 
9.660376|10.339624|| 9.635306 
9'660710) £0.339990|! 9.635570 


9.661043] 10,338957|| 9.655834 
9.661377 
9.661710 











t 





10.338290)} 9.636360 
10.337957|| 9.636623 
10.337624|| 9.636886 
10.337291)} 9.637148 


10.336958}| 9.637411 | 





9.662376 
sie 


9.663042 
9.663375 
9 663707 





10.336625|! 9.637673 
10.336293|| 9.637935 
9.664039] 10.335961|| 9.638197 
9,664.37 1'10.335629|| 9.638458 
9.664703] 10.335297; 9.638720 


9.665035) 10.334965); 9.638981 
9.665366|10.3346341| 9.639242 
9.665698) 10.334302}| 9.639503 
9.666029! 10.333971 9.639764 
oo ssa | 9.640024 














9.666691} 10.333309)| 9.640284. 


10.332979|| 9.640544 
10.532648)| 9.640804 
10.332318!| 9.641064 
10.331987|| 9.641324 
10.331657|| 9.641583 
10 331327|| 9.641842 


Tang. || Cosine. 





9.667352 





9.668013 
9.668345 
9.668673 











Sine. | Cosine. 








10.338623 638360 9.955005 














































Tang. | Cotang. 

. 9,66867 3|10.331327|60 
9 669002/10,330998159 
9,669332|10.330668)58 
9.669661 |10.330339|57 
9.669991|10,330009) 56 | 
9,670320|10.329680)55 
9.670649] 10,329351|54 


9.670977 |10.329023153 
9.671306|10.328694|52 
9.671635 |10.328365|51 
9.671963|10.328037|50 
9.672291 |10.327709|49 
9.672619] 10. 32738) |48 


. 9.67 2947}10.327053}47 | 
9.956447| 9.673274)10.326726|46 

9.956387; 9.673602) {0.326398 |45 
9.956327] 9.673929] 10.326071 |44 
9,956268| 9.674257|10.325743/43 
metal 9.6745 84|10.325416|42 


9.956148! 9.674911|10.325089! 41 
9.956089| 9.675237|10.324763|40 
9.946029] 9.675564|10.324436|38 
9.955969} 9:675890|10.3241 10|38 
ssi 9.676217|10.323783137 





9.957276 
9.957217 
9.957158 
9.957099 
9.957040 
9.956981 
9.956921 


9,956862 
9.956803 
9.956744 
9.95 028 





9.956625 
9.956566 


9.956506 











9.955849] 9.676543/10.323457 |36 


9.676869} 10,3231 31 (35 
9,677 194) 10.322806|34 
9.677520/10,322480} 93 
9.677846|10,$22154.132 

- 9.678171|10.321829131 

9.955488) 9.6784.96|10.32: 504/30 


9.955428, 9.678821 |10.321179|29 
9.955368] 9.6791 46|10.320854/28 
9.955307; 9.679471 10.320529/27 | 
9.955247! 9.679795] 10.320205 |26 
9.955186} 9.680120} 10.319880/25 
9.955126| 9.680444|£0.319556/24 


7s5suns 9.680768/10.319232|23 


9.955789 
9.935729 
9.955669 
' 9.955609! 
9.955548 








9.681092'10.318908|22 
9.954944! 9.681416/10.318584/21 
9.954883] 9.681740]10,318260!20 
9.954823) 9.682063/10,317937|19} 
9.954762) 9.682387 15.317615/18 


9.954701| 9.6827 10/10.317290| 17 
9.954640} 9.683033/10.316967 | 16 
9.954579] 9.683356) 10.316644, 15 
9.954518] 9.683679} 10.316321 't4 
9.954457| 9.684001|10.315999|13 
9,954396| 9.684524. 10.315676)12 i 


9.954335] 9.684646) 10.315354| EL 
9.954274; 9 684968/10,3) 503210 
9.954.213 9,685290)10.514710) 9 
9,954152| 9.685612/10.314388; 8 
9.954090, 9.685934|10,314066| 7 
9,954029| 9.686255)|10.313745) 6 


9.953968 9.686577 |10.313425| 5 
s0ss89 9.686898}10,313102| 4 





9.953845! 9. 687219|10,312781! 3 

9.953783| 9.687540|10.312460| 2 

9.953722| 9.687861|10.312139] 1 

9,953660} 9.685182/10.311818) 0 

Sine. | Cotan. | Tang. | ’ 
64 Deg. 


i) 








Dow HMoewiomo! ~ 


60 








Sine. : 
9.641842 
9.642101 
9.642360 
9.642618 
9.642877 
9.643135 
9.643393 


9.643650 
9.643908 
9.6441 65 
9.644423 
9.644680 
9.644936 


9.645193 
9,645450 
9.645706 
9,645962 
9.646218 
9.646474 


9.646729 
9.646984 
9.647240 
9.647494 
9.647749 
9.648004 


9.648258 
9,648512 
9.648766 
9.649020 
9.649274 
9.649527 


9.649781 
9.650034 
9.650287 | 
9.650539 
9.650792 
9.651044 


9.651297 
9.651549 
9.651800 
9.652052 
9.652304 
9.652555 


9.652806 
9.653057 
9.653308 
9.653558 
9.653808 
9.654059 


9.654309 
9.654558 
9,654808 
9,655058 
9.655307 
9.655556 


9.655805 


9.656799 
a. 657047 


| Cosine... 








Cosine. 


"9.953660 


9.953599 
9.9535 37 
9.953475 
9.953413 
9.953352 
9.953290 


9.953228 
9 953166 


9.953104 


9.953042 
9.952980 
9.952918 


9.952855 
9.952793 


9.952731 
9,952669 
9,952606 
9,952544 


9.952481 


9.952419 
9.952356 
9.952294 
9.9§2231 
9.952168 


9.952106 


9.952043 


9.951980 


9.951917 


9.951854 
9.951791 


9.951728 
9.951665 
9,951602 
9,951539 
9.951476 
9.951412 


9.951349 
9.951286 
9.951222 
9,951159 
9.951096 
9.951032 


9.950968 
9.950841 
G,950778 
9.950714 


9.950650 


9.950586 


9.950522) 9. 


9.950485 
9,950394 
9.950336 
9.950266 


9.656054, 9.959138 


Tang. 
9.688182 
9.688502 
9.688828 





9.656362} 9.950074, 
9.656561 9.950010) 9.706541 


LOG. SINES, TANGENTS, &c. 


eta a nr Aaa aan 


26 DO oe ae Te ener 


9.689143} . 
9.689463 
9.689783 
9.690103 


9.690425 
9,690742 
9.691062 
9.691381 
9.691700 
9.692019 


9.692338 
9.692656 
9,692975 
9.693293 
9.693612 
9.693930 


9.694248 
9.694566 
9.694883 


9.695201 
9.695518 
9.695836 


9.696153 
9.696470 
9.696787 
9.697103 
9.697420 
9.697736 


9.698053 
9.698369 
9.698685 
9.699001 
9.699316 
9.699632 


9.699947 
9.700263 
9.700578 
9.700893 
9.701208 
9.701523 


9,701837 


| 
9.7021 : 
| 
| 
| 
| 


9.702466 


9.702781 
9.703095 
9.703409 


9.703722 
704036 
9,704350 
9.704665 
9.704976 
9.705290 


9,705603 
9.705916 
9.706228 


9.949945) 9 706854 
9.949881) 9.707166 


~ Sine, 





63 Deg. 


“Gotan. 6 0 


_10,299107 


10 298163 


10.297219 





10.307025}- 


10.306707 
10.306388 
10,306070 


10.305752 
10.3054.34}. 
10,305117 
10.304799 


10.304482) 


10.304164 


{0.303847 
10.303530 
10.308213 
10.302897 
10.302580 
10.302264 


10.301 947 
10,301631 
10.301315 
10.300999} 
10.300684 
10,300368 


{0.300053 
10,299737 
10,299422 


10,.298792 
10,298477 


10,297848 
10,.297534 


10,296905 
10.296591 


10,296278 
10.295964. 
10.295650 
10,295 337) 
10.295024 
10294710 


10.294397 
10 294084 
10.293772 
10.293459 
10,293146 
10,299854 


Tang. 






“7 > Cotanme.-}| Sine:> | Gosine. ) Tanes | Cotane. 1~ 


9.657047 
9.657295 
9.657542 
9.657790 
9,658037 
9.668284 
9.658531 


9.658778 
9.659025 
9.659271 
9.659517 
9.659763 
9.660009 


9.660255 
9.660501 
9.660746 
9.660991 
9.661256 
9.661481 


9 661726 
9661970 
9.662214 
9.662459 
9.662703 
9.662946 


9.663190 
9.663433 
9.663677 
9.663920 
9.664163 
9.664406 


9.664648 
9.664891 
9.665133 
9.665375 
9.665617 
9.665859 


9.666100 
9.666342 
9.666583 
9.666824 
9.667065 
9.667305 


9.667546 
9.667786 
9,668027 
0.668267 
9.668506 
9.668746 


9.668986 
9.669225 
9.669464 


9.669703 


9,669942 
9.670181 


9.670419 
9.670658 
9.670896: 


. 9.671134) 9.946069 


9.671372 
9, 671609 


“Cosine. 





27 Bi sia 5% AAT ce ACRONIS RM Gay aan 


Cosine. 


“9.949881 
9.949816 
9.9497 52 
9.949688 
9.949623 
§,949558 
9, 949494 


Q, 949429 
9.949364. 
9.949300 


9.949235 


9,94.9170 
9.949105 


9,949040 
9.948975 
9.948910 
9.948845 
9.948780 
9.948715 


9.948650 


9.948584 
9.948519 


9.948454 


9.948388 
9.948323 


9.948257 
9.948192 
9.948126 
9.948060 
9.947995 
9.947929 


9.947863 
9.947797 
9.947731 
9.947665 
9.947 600 
9.947533 


9.947467 
9.947401 
9,947 335 
9.947269 
9.947203 
9.947136 


9.947070 
9.947004 
9.946937 
9.946871 
9.946804, 
9.946738 


9,946671 


9.946604}. 


9,946538 
9.946471 
9.946404 
9.946337 


9,946270 
9.946203 
9 946136 





9,94.6002 
9.945935 5 oa 


Sine. 
G2.Deg. 


9.712766) 10.287234) 42 | 


_ Tang. | Coiang. 


"9.7071 66| 10.292834|60 
9.707478) 10.292522)59 | 
9.707790} 10.292210158 | 1 
9.708102) 10,291898157| 
9.708414) 10.291586/56 | 
9.7 08726/10.291 274155 | 
9.709037}10.29096315 4 | 


9.709349] 10.29065 1153 | 
9,709660}10.29034.0159 | 
9.70997 1}10.29002915 1 | 
9.710282/10.289718150 | 
9.710593|10.289407|469 | 
9:7 10904 10. 289096 48 | 


9.7 711215 10.288785, 47 | 
9.711525}10.288475146 | 
9.711836/10.288164)45 | 
9.712146)10,287854)44 | 
9.712456) 10,287544143 | 



















9.713076'10,286924141 | 
9,713386|10.286614140 
9,715696|10,286304| 39 
9.714005|10,285995138 | 
9.71431 4|10,285686137 | 


9.714624)}10,285376136 


9.714933] 10.285067'35 | 
9.715242/10,.284758134 | 
9.715551|10,284449)33 | 
9.715860}10.284.140) 32 | 
9.7 1.6168) 10.283832138 ¢ 
9,716477|10.283523} 30 | 


0.716785]10.283215|29 | 
9.717093|10.282907/ 28 | 
9.717401110.282599|07 
9.717709|10.28220 1126 | 
9.7 18617|10.284983/95. 
9.718325 10,281675) 2 


9.718633|10.281 367123 
9,718940)10.281060122 
9.719248] 10.280752191 
9.719555|10.280445120° 
9,719862|10.280138\19 
9.720169}10.279831 |18 


9.72047 6|10.279524)17 | 
9.720783) 10.279217 161" 
9.721089|10.278911115 
9,.721396}10.27860411 4 | 
9.721702|10.278298113 
9.722009] 10.277994 \12 


9.722315|10,277685 14 
9.722621) 10.277379)10 
9.722927|10,277073] 9 
9.723232) 10.276768| 8 
9.723538)10.276462| 7 
9.723844'10,276156) 6 


9.724149)10.27585h) 5 
9.724454)10.275546) 4 
9.724760) 10.275240} 3 
9.725065|10,274935| 2 
9.72537 0)10,274630) 1 
9. 725674 10, 274326 OF 


- Cotan. i 

















Tan: e. 





< 


" LOG. SINES, ‘TANGENTS, &e. 








































28 

Sine. | Cosine. - 
9.671609} 9.945935 
9.671847} 9.945868 
9.672084! 9.945800]: 
9.672321). 9.955733 
9.672558] 9.945666 
9.672795! 9,945598 
9.673032] 9.945531 


9.675268} 9.945464 
e677 9.945396 


leg. poe: Ce 29 Deg. 
Cotang. |) Sine. | Cosine. | Tang. | Cotang. 


"10.274326|) 9.685571) 9.941819] 9. "9.743752|10.256248|60 
10.274021|| 9.685799} 9.941749] 9.744050110,255950159 
10.273716||'9.686027| 9.941679] 9.744348|10,255652|58 
10.273412!| 9,686254| 9.941609) 9.744645] 10.25535 5157 
10.273108|| 9.686482) 9.941539! 9.744943] 10.255057|56 
10.272803|| 9.686709) 9.941469] 9.745240]10,254760|55| 
10,272499]| 9.686936! 9.941398] 9.745538110.254462|54 


10.272195|| 9.687163} 9.941328] 9.745835|10.254165153 
10.271891}| 9.687389] 9.941258] 9.746132) 10,25 3868/52 
9.728412] 10.271588!| 9.687616] 9.941187} 9.746429] 10.253571|51 
9.728716] 10.271284. 9.687843} 9.941117 9.746726) 10.25527 4 50 
= 10.270980}| 9.688069} 9.941046} 9.747023} 10;252977 |49 
































Pang. 
9.725674 
9.725979 
9.726284 
9.726588' 
9.726892 
9.727197 
9.727501 


9.727805 
9.728109 





CON aurswcomol ~ 


9.673741] 9.945328 
9.673977| 9.945261 
9.674213} 9.945193 
9.674448) 9.945125 


674684 9.945058 
3 674919 9.944990 
9.675155| 9.944929] 
9.675390} 9.944854. 
9.675624! 9.944786 
9.675859} 9.944718 


9.676094! 9.944650 
saphena 9.944582 
9.676562! 9.944514 
9.676796) 9.944446 
9.677080) 9.944377 
9.677264) 9.944309 


677498] 9.944241 
ap ale 9.944172 
9.677964! 9.944104 
9.678197| 9.944036 
9.678430! 9.943967 
9.678663| 9.943899] 


9.678895) 9.943830 
9.679128! 9.943761 
9.679360| 9.943693 
say ta 
OF Le . 

0.680056! 9.943486 


9.680288} 9.943417 
9.680519! 9.943348 
9,680750| 9.943279 
9,680982} 9.943210 
9.681213} 9.943141 
9,681443} 9.943072 


9.681674) 9.943003 
9.681905! 9.942934 
9.682135 !-9.942864 
9,682365| 9.942795 
9.682595} 9.949726 
96828251 9.049656 


9.683055} 9.949587 
9.683284! 9.942517 
9 683514) 9.942448 
9.683743] 9.942378 
9.683972} 9.949308 
9.684201| 9.949239 


9,684430) 9.942169 
9.684658! 9.942099] 
9.684887] 9.942029 
| a6 l13) 9.941959) 9.745156 
{59} 9.685543) 9.941889] 9.743454 
| 90} 9.685571] 9.941819 9.743752 


9.729323} 10.270677|| 9.688295! 9.940975) 9.747319! 10.252681/48 


9,729626| 10.270374]| 9.688521! 9.940905] 9.747616] 10.252384|47 
9.729929] 10.270071/| 9.688747) 9.940834] 9.747913| 10.252087|46] 
9.730233] 10.260767/| 9.688972] 9.940763] 9.748209|10.251791|45 
9.730535} 10.269465|| 9.689198) 9.940693] 9.748505 10.251495]44 
9.730838] 10.2691691] 9.689423] 9.940622! 9.748801 '10,251199]|43 
9:731141| 10.268859)| 9.689648) 9.940551) 9.749097|10.250908|42 


9.731444] 10.268556)| 9.689873! 9.940480). 9.749393! 10.250607 |41 
9.731746| 10.268254)) 9.690098 9.940409) 9.749689|10.250311|40 
9.732048} 10.267952 260088 9.940338, 9.749985) 10.250015|39 












9.732351) 10.267649]| 9.690548) 9.940267, 9.750281)10.249719|38 
9.750576] 10,249424|37 
9.750872|10.249128| 36 


9.751167 /10,248833135 
9.751462) 10,248538|34 
9.751757 |10,24824.3|33 
9.752052) 10.247948|32 
9.752347 |10 24765331 
9.753642) 10.247358130 


9.752937 | 10.247 063|29} 
9.753231 |10.246769}28 
9.753526 |10.246474| 27 
9.753820) 10.2461 80} 26 
0.754115 | 


9.690772! 9.940196). 
9.690996] 9.940195 


9,6912201 9.940054 
9.691444| 9.939982 
9.691668} 9.939911 
Ea 9.939840) 


10.267347 
10.267045 


10.266743 
10.266442 
10.266140 
10.265838 
10.265537 
10.265236 


10.264934 
10.264633 
10.264332 
10.264031 
10.263731) 
10.263430 


10.263130 
10.262829 
10.262529 
10.262429 
10.261929 
10.26 1629 


10,261329 
10.261029 
10,260729 
10.260430 
10.266130 
10.259831 


10.259532 
10.259233 
10.258934, 
10.258635 
-10°258336 
10.258038} 


10.257739. 
10.95744} 
10,257149 
10.25684.4 
-10,256546! 
-10.256248 


9.752653 
9.732955 


9.733257 
9.733558 
9.733860) 
9.734162 
9.734463 
9.734764 


9.735066 
9.735567 
9.735668 
9.735969 
9.736269 
9.736570 


9.736870 
9.737171 
9.737471 
9.737771 
9.738071 
9.75837 1 
9.738671 
9.738971 
9.739271 
9.739570 
9.739870 
9.740169 


9.740468 
9.740767 
9.741066 
9.741865 
9.741664 
741962 
9.742261 


9.74.2559 
9,742858 





9.692115} 9.939768 
9.692339] 9.939697 


9.692562 9.939695 
9.6927 85| 9.939554. 
0.693008} 9.990482 
0.693231! 9.939410 
9.693453) 9.939330 
9 693676| 9.939267 


9.693898} 9.939195 
9,694120| 9.939123 
9,604342) 9.990052 
9.6945641 9.938980 
9,694780| 9.938908 
9,695007| 9.938836 


9.695229| 9.938763 
9.695450] 9.938691 
0.695671) 9.938619] 9 
9.495892! 9.938547 
9.696113| 9.938475 
9.696354) 9.958402 


9,696554| 9.938330 
9.696775} 9.938258 





9.754997 |10.245003|22 
9.755291 (0.244709|21 
9.755585) 10,244415 
0.755878|10.244129|19 
9, pine 10,243828 





J 557 Be eet 15 
4 757545} 10.242655|14 
9.757638]10.242362 
9.757931 |10.242069 








9.758517|10. ciMas 
9.696995} 9.938185] 9.758810}10,241 190] 
9.697215! 9.938113! 9.759102'10,240898} 
9.697435] 9.958040) 9.759305|10.240605 
9,697654! 9,93; al 9.759687 |10.240313 


9:697874) 9.937895) 9.759979]|10.240021 
9.698094, 9.937822 9.760272|10,239728 
9.698313] 9.957749) 9.760564, 10239436), 
9.6985: 52 9.937676] 9.760856|10.239144 
9.698751! 9.937604! 9.761148110. a 














eee 


ee an 


9.698970 9. 937531} 9, 761459/|10.238561 
“Sine. . | Gotan. 








_ Tang 


Cosine. 


be Cosine. ‘Ste. 


| ~ Cotan. 


Tang. 


60 Deg. 








reread 


Cosine. 


9.937531 
9.937458 

9.937385| 
9.937312 
9.937238 
9.937165 
9.937092 


9.937019 
9.936946 
9.936872 
9.936799 
9.936725 
9.936652 


oii 


Sine. 


| 9.698970 
9.699189 
9.699407 
9.699626 
9,699844 
9.700062 
| 9.700280 


9.700498 
| 9.700716) 
| 9.700933 


iz 














AN WWM O 


. 
nate era ek eee 22 pe 


aj 


8 


S 


10 
il 
12 


13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24, 


25 
26 
27 
28 
29 
# 


31 
32 
33 
34, 
35 
36 


9.701151 | 
9.701368 
BaOLEDH 


9.701802 
9.702019 
9.702236 
9.702452 
9.702669 
9.702885 


9.703101 
9.703317 


9.936505 
9.936431 
9.036357 
9.936284 
9.936210] 


9.936136! 
9.936062 
9.703533\ 9.935988 
9.703749| 9.935914 
9.703964| 9.935840 
9.704179] 9.935766 


9.704395| 9.935692 
9,704610| 9.935618 
9.704825] 9.935543| 
9.705040 9.935469 

705235 9.935395 | 
0.705469 9.935320] 


9.7: ose 9.935246 





2S a 





9.705898} 9.935171} 
9.706112| 9.935097 
9.706326) 9.935022 
9.706539) 9.934948 
9.706753} 9.934878 


9.706967| 9.934798 
9.707180! 9.934723 
9.707393) 9.934649 
9.707606) 9.934574 
9.707819) 9.934499 
708083 9.934424. 


9.708245) 9.934349 
9. Tos 9.934974 
9.708670] 9.034199 
9,708882| 9.934123 
47| 9, 709094) 9 9,934048| 
48| 9.709306) 9.933973 


49| 9.709518] 9.933898 
50| 9.709730} 9.933829 
51| 9.709941| 9.933747 
52| 9.710153| 9.933671 
53| 9.710364 
54| 9.710575 


551 9.710786 9.933445} 
4 9.710997; 9. 59009) 9 
57! 9, 711208 | Saeaaia 
io 9.711419 9,933217| 


9.933596), 
9,933520| 


59} 9.711629| 9.933141 
ie 9.711839 9.933066 


| Cosine. | Sine. 


Vor. Ui. 





Cotan. 


LOG, SINES, TANGENTS; &c. 


30 Deg. 


Tang _ Cotang. 


9.761439} 10.238561 


9.761731 )10.238269 








31 Deg. 


__ Sine. Cosine. 
9,711839) 9.933066 
9.712050} 9.932990 


9,.762023|10.237977)| 9.712260) 9.932914 
9.762314) 10.257686|| 9.712469) 9.932838 


9 762606} 10,237594, 


9.762897 | 10.237103 
9.763188! 10236812 


9.763479)10.236521 
9.763770| 10.236230 
9.764061 |10,235939 
9.764352) 10.235648 
9.764643 | 10.235357 





9.764933! 10.235067 


9.765224 '10.234776 

9.765514|10.234486 
9.765805|10.234195 
9.766095 |10.233905 
9.766385 |10,233615 
9,766675|10,233325 


9.766965 |10.233035 | 
9.767255 |10.232745 
9.767545] 10,239455 
9.767834) 10,232166 
9,768124|10,231876 
9, sina 10,231586 


9. 768703| 10.231297 
9.768992|10.231008 
9.769281 10.230719 
9.769571) 10.230429 
9.769860/10.230140 
9.770148) 10,229852 


9.770437 |10.229563 
9.770726 |10.229274 
9. 771015 |10. 228985 
9.771303) 10.228697 
9.771592\10.228408 
9.771880|10.228120 


9.772168 | 10.227832 
9.772457 |10.227543 
9.772745 |10.227255 
9.773033} 10.226967 
9.773321 |10.226679 
9.773608 | 10.226399 


9.773896|10.226104)' 
9.774184) 10. 225816) 
9.774971 10, eel 
9.774759, 10. 224084 | 
9.775046) 10. 224084 | 
9.775333) 10.224667 


9.775621 Lozeiar9| 
9.775908] 10,224092 
9.776195|10.223805 
9.776482|10,223518 
9.776769 | 10,223232 
9.777055)10.222945 


9.777342|10.222658 
9.777628) 





9.778201 |10.221799 
9.778488 | 10.221512 
9, 778774 10,221226 


Tang. 


10,222372' 
9.777915 110.222085 || 9.723608 




















i} 











| 











| 

















| 


9.712679) 9.932762 
9.712889) 9.932685 
9.713098! 9.932609 


9.713308; 9.932533 
9.713517) 9.932457 
9.713726) 9.952380 
9.713935| 9.932304 
9.714144} 9.932228 
9.714352! 9,932151 


9.714561| 9.932075 
9.714769| 9.931998 
9.714978} 9.931921 
9.715186) 9.931845 
9.715394] 9.931768 
9.715602) 9.931691 


9.715809! 9.951614 
9.716017| 9.931537 
9.716224) 9.931460 
9.931383 
9,931306 
9.931929 


9.931152 
9,931075 
9.930998 
9.930921 
9.930843 
9.930766 


9.930688 
9.930611 
9.930533 
9.930456 
9.930378 
9.930300 


9.930223 
9.930145 
9.930067 
9.929989 
9.929911 
9.929833 


9.929755 
9.929677 
9,929599 








9, 716639 
9.716846 


9.717053 
9.717259 
9.717466 
9.717673 
9.717879 
9.718085 


9.718291 
9.718497 
9.718703 

9.718909 
9.719114 
9.719320 


9.711525 
9.711730 
9.71 1935| 
9.720140 
9.720345 
9.720549 


9.720754 
9.720958 
9.721162 
9.721366) 9.929521 
9.721570) 9.929442 
Lea 9.929364 


9.721978} 9.929286 
9,722181| 9,929207 
9.722385) 9 929129 
9.722588} 9.929050 
9.722791| 9.928972 
9.722994) 9.928853 


9.723197 
9.723400 





9.723805 
9,724007 
9.724210 


‘Cosine. 


9.928578 
9.928499 
9.928420 


Sine. _ 








9.998815! 9.794383110.205617 
9.928736! 9.794684 
9.928657| 9.794946 
9.795927 





Tang. | Cotang. a 
9.77 8774) 10.221226|60 
9.779060) 10,220940159 
9.779346] 10,22654|58 
9.779652) 10.220368] 57 
9.77991 8} 10.220082)| 56 
9.780208] 10,219797|55 
9.780489) 10.219511|54 


9,780775| 10.21 9225 |53 
9.781060! 10.21894.0|52 
9.781346|10.218654/51 
9.781631 |10.218369|50 
9.781916|10.218084/49 
9.782201) 10.217799|48 


9.782486) 10.217514|47 
9,78277 1 | 10.217229|46 
9.783056|10.21694.4.|45 
9.783341) 10.216659 144 
9.788626) 10.216374\49 
9.783910) 10,21 6090/42 


9.784195) 10.215805 |41 
9.784.479] 10.215521 |40 
9.7847 64 10,215236|39 
9.785048 | 10.214952/38| 
9.785332) 10.214668)97 | 
9.785616) 10.214384/36} 


9.785900}10.214100'35| 
9.786184) 10.213816|34! 
9.786468 |10.213532|33} 
9.786752|10.213248/32 
9.7 87036) 10.212964'51} 
9.787 319/10,219681|90} 


9.787603!10,212397|29 
9.787886|10.212114|28| 
9.788170) 10.211830|27 | 
9.788453) 10.211547/26} 
9.7887 36| 10,211 264/25} 
9.788019} 10.210981 |24: 


9.789302| 10.21 0698/23} 
9,789585/10.210415)22| 
9.789868] 1 0.9101 32/21} 
9.790151] 10,.209849/20} 
9.790434, 10.209566!19} 
9.790716) 10,209284)18 


9.790999] 10.209001|17| 
9.79128 1|10.2087 1916} 
9.791563} 10.2084.37| 15} 
9.791846] 10.2081 54/14) 
9.7921 28} 10.207872)13 
9.792410) 10,.207590)£2 


9.792692) 10, 207308)11) 
9.79297 4, 10,207 026) 10} 
9.793256} 10206744, 
9.793538/10,206462 
9.793819 110.2061 81} 
9.794101 |10.205899 

















10.205054 
10.204773 
9.795508 '10.204492| | 
9.795789] 10.204.211 


——— | — fh. 


“Tang. 


to 205336 


TGothna,, 





Con Gaeworol 








> 9.725823 


oe 


Sine, _ 


9.724210 
9.724412 


9.724614 
9.724816 
9.725017 
OT LIZY 
9.725420 


9.725622 


9.726024 
9.726225 
9.726426 
9.726626 


9.726827 
9.727027 
9.727 228 
9.727428 
9.727628 
9.727 828 


Cosine. | _ 





9.728027 
9.728227 
9.728427 


9.728626 


9.728825 
9.729024 


9.729223 
9.729422 
9.729621 
9.729820 
9.730018 
9.730217 


9.730415 





9.730613) 
9.730811 


g. -731009)| 9.925707 


9.731203 
9.731404 


9.731 602) 
9.731799 
9.731996 
9.732193 
9.732390 
9.732587 


9.732784 
9.732980 
9.733177 

33373 
9.733569 
9.733765 


9.733961 
9.734157 
9.734353 
9.734549 
9.734744 
9.734939 


9.735135 
9.735530 





9.735525} 


9.735719 


9.735914 


9.736109 


‘Cosine. ' 


See 


Ce et 





9.928420 


9.928342; 


9 928263 
9.928185 
9.928104 
9.928025 
9.927946 


9,927867 
9.997787 
9.927708 
9.927629 
9.927549 
9.927470} 


9.927390 
9.927310 
9927231 
9.927151 
9.927071 
9.926991 


9.926911! 
9.926831 
9.926751 
9.926671 
9.926591 
9.926511 


9.926431 
9.926351 
9.926270 
9.926190 
9.926110 
9.926029 


9.925949 
9.925868 
9.925788 








9.925626 
9.925545 
9.925465 
9.925584 




















LOG. SINES, TANGENTS, &c. 


32 32 Deg, 
Tang.” 


9.795789)|10.204211 
9.796070/10.203930 
9.796351 |10.203649 
9.796632) 10.203368 
9.796913)10,203087 
9.797194) 10.202806 


9.797474)10,202526} 


Cotang. - 


9,797755}10.202245 
9.798036/10.20!1 964 
9.798316)|10.201684 
9.798596}10.201404 
9.798877 | 10.201 123 
9.799157 |10,200843 


9.799437 
9.799717 
9.799997 


9.800277 


9.800557 
9.800836 


9.801116 


9.801396) 
9.801675, 


9.801955 
9.802254 
9.802513 


9:802792 
9.803072 
9.803351 
9.803630 
9.803909 
9.804187 


9.804466 
9.804745 
9.805023 
9,8065502 
9.805580 
9.805859 


9.806137 
9.806415 





10.200563 
10,200283 
10.200003 
10.199723 
10.199445 
10.199164 


10.198884 
10.198604 
10.198325 
10.198045 
10,197766 


10.197487| 


10,197208 
10.196928 
10.196649 
10.196370 
10.196091 
10.195813 


10.195534. 
10,195255 
10,1 94977 
10.194698 
10.194420 
10.194141 


10.193863 
10,193585 


9.925303! 9.806693/10.193507 
9.925222) 9.806971 |10.193029 
9.9251411 9.807249) 10.192751 
9.925060} 9.807521 /10.192473 


9.924979) 9.807805 |10.192195 
9.924897) 9.808083 |10.191917 
9.924816} 9.808361)10.191639 
9.924735} 9,808638/10.191362 
9;924654| 9.80891 6)10.191084 
9.924572! 9.809193) 10.190807 


9.923591 





9.924491 | 
9.924409 


9.924083 


9.924001 | 
9.923919, 

9.923837 
9.923755 
9.923673 








Sine. 





9.80947 1}10.190529 
9.809748] 10,190252 
9.924328) 9.810025) 10,189975 
9.924.246! 9.810302! 10.189698 
9.810580} 10,189420 
9.810857/10,189143 


9.811134) 10. 188866) 
9.811410] 10.188590 9.746812 


9.811687/10.188313 
9.811964)10,188036 
9.812241 /10.187759 


9.812517] 10.187485 


Cotan. — 
_57 Deg. 


eases 


| 


Tang. 





Sine. - | Cosn 
"0.7361 109. 
9, 736303' 
9.736498 

9.736692; 
9.736886! 
9.737080 
9.737274 


9.737467 
9.737661 
9.787855 
9.738048 
9.738241 
9.738434 


9.738697 
9.738820 
9.739013 
9.739206 
9.739398 
9.739590 


9.739783 
9.739975 
9.7401 67 
9.740359 
9.740550 
9.740742 


9.740934 
9.741125 
9.741316 
9 741508 
9.741699 
9.741889 


9.742080 
9.742271 
9.742462 
9.742652 
9.742842 
9.743033 


9.743223 
9.74345 3 
9.743602 
9.743792 
9.743982 
9.744171 


9.744.361 
9.744550 
9.744739 
9.744928 
9.745117 
9.744306 


9.745494 
9.745683 
9.745871 
9.746060} 
9.746248 
9.746456 


9.746624 























9.746999 
9.747187 
9.747374 
9.747562 


Cosine. 








33 Deg. 


Cosine. 


9.923591 
9.923509 
9.923427 
9.923345 
9.923263 
9.923181 
9.923098 


9.923016 
9.922933 
9.922851 
9.922768 
9.922686 
9.922608 


9 922520 
9.922438 
9.922355 
9.929972 
9.922189 
9.922106 


9.922023 
9.921940 
9.921857 
9.921774 
9.921691 
9.921607 


9.921524. 
9.92144) 
9.921357 
9.921974 
9 921190 
9.921107 


9.921023 
9.920939 
9.920856 
9.920772 
9.920688 
9.920604 


9.920520 
9.920436 
9.920352 
9.920268 
9.9201 84 
9.920099 


9.920015) 
9.919931] 
9.919846) 
9.919769 
9.919677 
9.919593 


9.919508 
9.919424, 
9.919339 
9.919254 
9.919169 
9.919085 


9.919000 
9.918915 
9.93 8831 
9.9: 8745 
9.918659 


9.918574 


Sine. 


Cotan.- 
56 Deg. 


~~ 
T ang. Cotang. | 
9.812517\10.187483 60! 
9.812794) 10.187206)59 
9.813070) 10.186930/58 
9.813347 | 10.186653/57 
9.813623|10.186377 |56} 
9.815899) 10.186101 |55} 
9.814176) 10.185824 154 


9.814452) 10.185548 |53 
9.814728) 10,185272)52 
9.815004) 10,184996)/54 
9.815280) 10.184720|50: 
9.815555 |10.184445 |49) 
9.815831) 10.184169)48 


9.816107} 10.183893|47 
9.816382) 10.183618)/46 
9.816658] 10.183342}45 
9.816933] 10183067 |44 
9.817209} 10.1827 91/45 
9.817484, 10.182516/42} 


9.817759] 10.1 82241} 41 
9.818035]10.181965|40 
9.818310] 10.181690|39 
9. 818585110.181415|38 
9.818860} 10.181140}37 
9.819135] 10.180865 |36 


9.819410] 10.180590|35 
9,819684/10.180316|/54 
9.819959) 10.180041/33 
9 $20234)10.179766 |32 
9 820508} 10.179492/3L 
9.820783] 10.179217/30 


9,821057|10.17894.35|29 
9.821532) 10.178668)/28 
9.821606} 10,178394|27 
9.821880] L0.178120/26 
9,8221 54) 10.177846) 25 
9.822429] 10.177571|24 


9.822703} 10.177297 23 
9.822977 | 10.177023)/22 





9.823251) 10.176749/21 
20}. > 


9.823524} 10.176476 
9.823798} 10.176202)19) 
9.824072} 10.175928/18 


9.824.545} 10.175655|17 
9.824619} 10.175381 {16 
9.824893) 10,175107 15 
9.825166) 10.174834)14 
9.825439] 10,174561/ 13 
9.825713) 10.174287)12 









9.825986) 10.17 4014111 


9.826259} 10.173741|10 
9.826532! 10.173468 
9.826805] 10.173195 
9.82707 8} 10172922 
9.827351) 10.172649 


9.827624! 10.172376 
9.827897 |10.17210 
9.828 70]10.171830 
9,828442|1 0.171558] 
9.828715)|10,171285 
9.828987 |10.171013 


Tang. 


SlOomim men aud]: 








we ene eer, 





DOM Are WI RO! > 








157 
(58 
59| 
60} 9.75 


4 


es 


9.748083. 


9.753312 


(56) 





LOG. SINES, TANGENTS, &c. 


34 Deg 


sine, Cosine. 
9.747562) 9.918574 
9.747749: 9.918489 
9.747936)| 9.918404 
9.748123, 9.918318 


9.748310: 9.918233 


9.748497 9.918147 
9.918062 


9,748870' 9.917976 
9.74056, 9.917891 
9.749243 9 917805 
9.749429, 9.917719 
9.749615) 9 917634 
9.749801) 9.917548 


9. 749987 9.917462 
9.750172! 9.917376 
9.750358 9.917290 
9.750543, 9:917204 
9 (750729 9.917118 
9.750914 9.917032 


ar7st099! 9.916946 
9.916859 


9.751284: 

9,751469! 9.916773 

9.751654, 9.916687 
9.916600 


9.751839) 
9.752023; 9.916514 
9.916427 


9.752208) 
9.752392' 9,916 341 
9.752576'.9.91 6254 
9.752760' 9.916167 
9.752944! 9.916081 
9.753128 | 9 915994 

9.915907 


9.915820 
9.915733 
9.915646 
9.915559 
9.754229 9.915472 


9.754412! 9,9145385 
9 754595, 9.915297 
9.754778 9.915210 
9.754960) 9.915123 
9.755143) 9.915035 


9.753495, 
9.753679 
9.753862 
9.754046 


o| 9.755326) 9.914948 


9.755508; 9 914860 


9.755690} 9.914773}, 


9. 755872: 9.914685 


9... 56054, 9.914598 


9. 756236, 9.914510 
9 756418 | 9.914422 


9.756600} 9.914334 
9.756782! 9.914246 
9.756963, 9.914158 
9.757144. 9.914070 
9.757326, 9.913982! 


9.757507| 9. aise 


9 757688| 9.913806 
9.757869, 9.913718 
9.758050. 9.913630) 
9.758230 9.913541) 
9.758411) 9.913453 
9.758591| 9.913365 


“Cosine. Sine. 








55 Deg. 
ew tT Aes: 


T: ang. Votang. 
9.828987 | LO. 0.171015 


9.829260] 0.170740} | 9.758772 
{0.170468 9.758952 


9.829532 
9.829805 
9,830077 
9.830349 
9.830621 


9,830893 
9.831165 
4.831437 
9,831709 
9.831981 
9.832253 


9.832525 


10170595 
10.1699:235 
10.16965% 
10,169379 


10.169107 
10.168835 
10.168563 
10.168291 
10.168019 
10.467747 


10:167475 
9,832796|10.167204 
9.833068} 10.166932 
9,833339)10.165661 
9.833611) 10.166389 
9.833882]10.166118 


9.834154; 10165846 
9,834425}10.165575 
9.834696] 10.165304 
9,834967|10.165033 
9.835238] 10,164762 
9 835509} 10.164491 


9.835780|10.1642 20 
9.836051 |10.163949 
9.836322|10, 163678 
9,836593)|10. 165407 
9,836864/10.163136 
9 837134|10.162866 


9.837405 |10.162595 
9.837675|10.162325 
9.837946 |10.1 62054 
9 $38:210|10.161754 
9.838487 10.1615 13 
9 838757|10.161243 


9.839027 |10.160973 
9.839297 10.160703 
9.839568 10.160432 
9.839938| 10.1601 62 
9.840108) 10.159892 
tind 10.159622 


9,84.0648/10.159352 
9.840917; 10.159083) 
9.841187] 10.158813 
9,841457|10.158543 
9.841727 |10.15827 3 
9,841996| 10.158004 


9.842266) 10.157734 
9.842535|10.157465 
9.842805) 10.157195 
9.84307 4) 10.156926 
9.845343) 10.156657 
9.843612, 10.156388 


| 9.843882)10.156118 
9.844151 /10.155849 





9.844.4:20]10.155580) 


| 9.844689/10.155311 
9.844958) 10.155042 
9.$45227 | 10,154773 


“Gotan. | Tang. 





-9.759312 


. 9.760031 


9762177 


- 9.762889 





































35 Deg. wala 
Sine. | Cosine. | ‘Tang. 1 Cotang. 
: 9.758591| 9.913365! 9,845227|10.154776 
9, 913279, 9.845496) 10.154504 
9,94 3187) 9.845764, 10,154236 


9.759% 32) 9.913099 
9.913010 
9912922) 


9.912833 


9.912744 
9.912655 
9,912566 
9.9: 94.77 


9.759492 
9.759672 


9.759852 


9.760211 
9.760390 
9.760569 
9.760748 


9.760927 
9.761106 
9.761285 
9.761464 
9.761642 
9.761821 


9.761999 





9.912299 


9.912210 
9.912121 
9.912031 
9.911942 
9.951853 
9.911763 


9:91 1674 
9.911584 
9.911495 
9.911405 
9.911315 
9.911226 


9.911136 
9.911046 
9.910956 
9.910866 
9.910776 
9.910686 


9.910596 
9.910506 
9.910415 
9.950325 
9.910235 
9.910144 


9.90054 
9.909963 
9.909873 
9,9097 82 
9.909691 
9.909601 


9.909510 
9.909419 
9.909328 
9 909237 
9 909146 
9.909055 


9.908964 
9.908873 
9.908781 
9908690 
9.908599 
9,908507 


9.908416 
9,768522| 9.908324 
9.768697] 9 908233 
9.768871; 9.908141 
9.769045; 9.908049 
9.769219 


Cosine. 


9.762356 
9.762534 
9,762712 


9.763067 
9.765 1245 
9.763422 
9.763600} 
9.763777 
9.763954 


9.764131 
9.764308 
9.764485 
9.764662 
9.764838 
9.765015 


9.765191 
9.765367 
9.765544 
9.765720 
9.765896 
9.766074 


9.766247 
9.766423 
9.766598 
9.766774 
9.766949 
9.767124 


9.767300 
9.767475 
9.767649 
9.767824 
9.767999 
9.768175 


9.768348 








Sine. 


9.919388! 





9.907958 pa 


9.846033) 10.153967 
9,8.6302' 10,153698, 
9.846570 10.153430, 
9,846839| 10.153161 


9.847 108 
9. 847 32%6!10.152624. 
9, 847644/10. 152356 
9.847913! 10.152087 
9.848181.10.15. 819 
9.848449/10,151551 


9 848717|10.151283 
9.848986}10.151014 
9,849254/10.150746 














60 
59 
58 
57 
56 
55 
54 


10.152892)53 


52 
ot 
50 
49 
48 
47 
46 
45 


> 


9.849522)10,15047 8/44 


9.849790) 10.1502i0 
9.850057|10.149943 


9,850325|10.149675 
9,850593|10.149407 
9.850861 |10.149139 


43 
42 


“i 
40 
39 





9,851129|10.148871 
9.851396)10.148604 
9.851664|10.148336 


9,851931/10.148069 
9,852199|10,147801 


38 
37 
36 


35 
34 





9.852466110,1475 34. 
9,852733|10,147267 
9.853001} 10,146999 
9.853268 10.146732 


9,853535/10,146465 
9.853802) 10.146‘ 98 


33 
32 
31 
30 
29 
28 





SE SS ee 


9.854069} £0.145931/27 | 


9,854336/10,145664 
9.854603}10.145397 
9.854870) 10.145130 


9,855137/10,144863" 


9.85540410.144.596: 


26 


25 


24 


23 
22 


9,855671/10.144529|21 
9,855938)10.144062 20 


9.856204: 10,145796 
9.856471 |10,14.3529 


9.856737|10.143263 
9:857004 10,142996 
9.857270|10. 142730 
9.857537|10, 142463 
9.857803)10,142197 
9.858069/10.141931 


9.858336) 10.141664 


On 


i8 


17 
16 
15 
14 


13 
12 


11 





9.858602] 10,141398 
9.858868) 10.141132 
9.859134/10.140866 
9.859400 
9,859666)10,143340 


9.859932/10.140068 
9.860198110, 139802 
9.860464) 10.139536 
9.860730) 10.139270 
9.860995)10.139005 
9.861261/10.138759 


Cotan. _ Tang. 





54 Deg. 


pore nee te 


10,140600, 


10 


ge MIMO 


Mons 


a a ee i re Be ge NR RN 


LOG. SINES, TANGENTS, &c. 


36 Deg. 


Sine. Cosine. 


9.769219) 9.907958 

ae 9.907866 
.769566| 9.907774 

9, 769740) 9.907682 
9. "769913| 9.907590) 
9.770087 | 9.907498 
9.770260} 9.907406 


9.770433| 9.907314, 
9.770606} 9.907222 
9.770779; 9.907129 
9.770952! 9.907037 
9.771125) 9.906945 
2} 9.771298) 9.906852 


9.771470) 9.906760 
9, 771643) 9.906667 
9.771815! 9.906575 
9.771987] 9.906482 
9.772159| 9.906389 
9.772331| 9.906296 


9.772503) 9.906204 
9.772675) 9.906111 
9.772847) 9.906018 
9.773018] 9.905925 
9, 773190 9.905832 
9. hated 9.905739 


9.773533} 9.905645 
9.773704) 9.905552 
9.773875) 9.905459 
9.774046| 9.905366 
9.774217} 9.905272 
a; tl 9.905179 


= 
Of CON LAOS = 


= 
5 = 





= —_ pa 
to 3S iS oe 34 ncn et ke 





to 
to 


23 


LS?) 
PSS 


rt por 
NED or 


28 


9.774558] 9.905085 
9.774729; 9.904992 
9.774899! 9.904898 
9.775079) 9.904804 
9.775240! 9.904711 
9.775410} .9,904617 


9.775580) 9.904523 
9.904429 
9.904335 
9.904.241 
9.904147 
9.904053 


9.776598} 9.903959 
9.776768! 9.903864 
9.776937| 9.903770 
9.777106| 9.903676 
9, 777275, 9.903581 
9, cabs 9.903487 


9.777613) 9.903392 
9.777811 9.903598 
9.777950| 9.903203 
9.778119) 9.903108 
9.778287} 9.903014 
9.778455! 9.902919 


9.778624) 
9.787921 9.909729 
9.778960| 9.902634 
pb g7 9128) 9.902558 

9.779295; 9.909444 
8 9. 779463| 9.902349 


| eT i ee eee oe Cosine. | Sine 


Go oto 
om OO 


2 Gd G2 Oo 
On ibe OO 


36 


we 
“J 


9.775759 
9.775920 
9.776090 
9.776259 
9.776429 


iB 1B B Co Go & 
wre COM 


i 





i Ne tee 





 9.870529110. 129471 


9.902824) 


Tang. | Cotang. 


9.861261|10.138739 
9.861527|10,138473) 
9.861792|10.138208 
9.862058'10.137942, 
9.862323] 0.137677 
9,862589|10.137411 
9.86:2854|10, 137146 


9.865119/10.136881 
9.863385) 10.136615 
9.863650]10,136350 
9.863915) 10.156085 
9.864180) 10.135820 
9.864445)10.135555 


9,8647 10) 10.135290 
9.864975) 10.135025 
9.865240! 10,134760 
9.865505! 10,1344.95 
9.865770) 10,134230 
9.866035/10.133965 


9.866300! 10.133700 
9.866564/10,153436 
9.866829) 10,133171 
9) 867094 10.132906 


i) 

















9, 867623 10. 132377 


9.867887/10.132113 
9.868152/10.131848 
9.868416/10.131584 
9.868680/10.131320 
9.868945 }10,131055 
9.869209] 10.130791 


9,869473/10.130527 
9.869737 |10.130263 
9.870001 }10,129999) |- 
9.870265)10.129735 





9.870793) 10.129207 


10. 128945) 
19.128679 
10.128415 
10.128151 

10,127888 
9.87237 6\10.127624 


9.872640) 0.127360 
9.872903) 10.127097 
9.873167'|10 126853 
9.873430) 10.126570 
9.873694 10.126306 
9.873957 |10.126043 


9.874220/10.125780 
9.874484) 10.1 25516 
9.874747)10.125253 
9.8750%0)10.124990) 
9.875273) 10°124797 
9.875537)10. ela 


9.875800) 10.124200 
9.876063) 10.123937 
9.876326) 10.123674 
9.876589) 10.123411 
9.876852)10,123148' 
9. 877114 10.122886 


“Gotan. 


9.871057 
9.871321 
9.871585 
9.871849 
9.872112 








“Tang. | 


53 Deg. 


ig ena oe 


Cosine. - 





37 Dee. 
Cosine 
9.902349) 
9.902253 























ang, | Cotang - bes 
9.877114) 10,122886|60 
9.877377} 10,122623|59 


Sine. 


“9.779463 
9.779631 
9.779798 
9.779966 
9.780133 
9.7 80300 
9.7 780467 


9.780054 
9.780801 
‘9.780968 
9.781134 
9.781301 
9.7 81468 


9.781634 
9.781800 
9.781966 
9.782132 
9.782298 
9.7 frees 


9.782630 
9.782796 ‘ 
9.782961 9.900337) 9.882625 |10,117375|39 
9.783127! 9. 900240) 9,882887/10.117113|38 
9.785292) 9.960144! 9.883148] 10,116852|3 

9.783458) 9.900047} 9.883410/10,116590|36 


9.783623 9.899951! 9.883672110,116328|35 
9.783788) 9.899854| 9.883934'10.116066|34 
9.783953] 9.899757| 9.884196/10,115304|33 
9.784118} 9.899660; 9.884457) 10.115543| 32 
9.784082 9.899564 9.884719 10.115281 31 
9.784447| 9,899467| 9.884980! 10,115020|30 


9.784612) 9.899370] 9.885242! 10,114758|29 
9. Te 


9.902158! 9.877640) 10.122360|58 
9,902063| 9.877903] 10.122097|57 
9.901967| 9.8781 65|10.121835) 56 
9.901872} 9.878428] 10.121572|55 
9 avi774 9,878691110,121309|54 


9.901681; 9.878953} 10.121047|53 
§.901585| 9.879216'10.120784}52 
9.901490) 9.879478) 10.120522)51 
9. 901394, 9.879744 |10.120259|50 
9,901298; 9.880003) 10.11.9997 |49 
9. "9012021 9. "88026510. 119735 |48 


9.901106! 9.880528! 10.119472|47 
9.901010! 9.880790] 10.119210146 
9.900914} 9.881052/10.118948}45 
9, 900813) 9,881314'10.118686|44 
9.900722! 9.881577 110.118423 

9,900626| 9.881839] 10.118161 142 


9.900529! 9.882101!10.117899/41 





= 





0.784941 
9.785105} 9.899078} 9.886026 
9.785269| 9.8989811 9.886288] 10.113719|25 


9.785433 9.898884) 9.886519|10.113451|24 


9.785597| 9.898787| 9.886811110.113189|23 
9.785761| 9,898689| 9.887072/10.112928] 22 
9,785925| 9.898592) 9.887393) 10.112667| 21). 
9.786089) 9. oor 9.887594] 10.112406)20} 





9,786252| 9.898397|.9.887855|10.112145|19] 
9.786416) 9.898299 9,888116|10.111884]18 


9.786579) 9.898209| 9.888378|10,1116291 17 
9.786742] 9.898104) 9.883639/10.111361 
9.786906] 9.898006, 9.888900] 0.111100 
9.787069] 9.897908! 9.889161 /10.110839 
9.787232) 9.897810) 9.889421 |10.110579 
9,787395| 9. HK 9.889682/10.110318 











9.787557] 9.897614) 9.889943| 10.110057 
9.897516) 9.890204] 10.109796 
9.897418 8 890465) 10109535 
9.897320! 9.890725) 10.109275 
9.897222| 9.890986|10.109014], 
9.897123! 9,891247/10.108753 


9.897025) 9.891507|10.108493 
9.896926) 9, 20980. 108232 


9.787720 
9.787883 
9.788045 
9.788208 
9. 788370 


9.788532 
9.788694 
9.788856 
9.789018 
9.789180 
9.789342 





9.896828) 9.892028|10.107972 
9.896729) 9.892289110,1077 15 
9.896631} 9.892549|10.107451 
9. 396532| he 892810/10.107190 


“Sine. | Cotan. _ 
52 Deg. 


ma tl OS Bor GNI 





Tang. 








| ecermenee ne neers 


Sine. 
9.789342 
9.789504 
9.789645 
9.789827 

‘| 9.789988 
9.790149 
9.790310 


9.790471 
9.790632 
9.790793 
9.790954. 
9.791115 
9.791975 
9.791436 
9.791596 
9.791757 
| 9.791917 
9.792077 
9.792237 


9.792307 
9.792557 
9, 792716) 
22| 9.799876 
23| 9.793035! 
24! -9,793195 


25| 9.793354. 
(26) 9.793514 
27| 9.793673! 
28| 9.793833) 
29! 9.793991 
. 9.794150) 


31} 9.794308 
32, 9.794467 
33! “9.794626 
34| 9.794784! 
9.794942! 
9.795101 


9.795259 
9795417 
9.795575 
9.795733 
9.795891 
9.796049 


9.796206 
9.796364 
9.796521 
9.796679 
7} 9796836 
9.796993 


9.797150 
9.797307 
| 9.797464 
0.797621 
9.797777 
9.797934 


9.798091 
9,798247 
9.798403 
9.798560 
9.798716 
9,7 cee 


aa “ 
SOwoon Ouswomol > 


Il 


21) 

















He WB PS 
He 09 a 














‘1 Cosine. 





LOG. SINES, TANGENTS, &c. 





Tang. ang. 


9.892810)10. 10.107190 
9.893070)10. 106930 
9. 893331) 10,106669 
9.893591; 'L0,106409 
9.89385 1| 10.106149 
9.8944 11/10, 105889 
9.894.372110.105628 


9.894632! 10.105368 
9.804891 |10.105108 
9.895152|10.104848 
9,8954.12/10.104588 
9.895672) 10.104328 
9.895932|10 101068 


9,896192|10,103808 
9.896452|10 103548 
9,896712|10.103288 
9.896971 |10.103029 
9.897231 |10.102769 
9.897491 |10,102509 


9.897751) 10.102249 
9,898010|10.101990 
9.898270/10.101730 
9.898530) 10.101470 
9.898789} 10.101211 
9.899049)10.100951 


9.899308! 10.100692 
9.899568|10,100432 
9,899827 
9.90008711 0.099913 

9,900346|10.099654 
9.900605 | 10.099395 


9,900864 10,0991 36 
9,90 11:24|10,098876 
9.901383] 10.098617 
9.901642) 10.098358 
9.901901 |10.098099 
9,902165|10,097840 


9,902420| 10,097580 
9.902679|10.097321 
9.902938) 10.097062 
9.903197 |10.09680,3 
9.903456110.096544. 
9.9037 14|10.096286)| 


9.892233) 9.903973|10.09027 
9.892132 9.904232! 10.095768 
9.892030! 9.904491 (0.095509 
9,891929, 9.904750/10.095250 
9,894 827; 9.995008)10,004992 

9.905267110,094733 


9.891726 
9.891624 9.905525) 10.094474 
9.891523 9.905785! 10.094215 
9.894421, 9.906043] 10,093957 
9.891519. 9,906:302|10.093698 

9.908560 )10 093440 


9.891217) 
9.801115! 9.906819119.093181 


9.891013} 9.907077}10,092923 
9.890911! 9.907336|10,092664 
9,890809 9.907594|10,092406 
9.890707| 9.907853) 10.0921 47| 
9.890605} 9.908111) 10.091 889 
9.290503} 9.908369/10.091631 


Sine. | Gotan. | Tang 


38 Deg. 
_| Cot Cotang. 


9.896532} 9 
9.896433 
9.896335 
9 896236 
9.896137 
9.896038 
9,895939 


9.895840 
9.895741 
9.895641 
9.895542 
9.895443 
9.895343 


9.895244 
9.895145 
9.895045 
9.894945 
9.894846) 
9.894.746] 


9.894646 
§.894546 
9.894446 
9.894546 
9.864246 
9.894146 


9.894046 
9.893946 
9,893846 
9.893745 
9.893645 
9.893544. 


9.893444, 
9.893343 
9.893243 
9.893142 
9.895041 
9.892940 


9.892839 
9.892739 
9.892638 
9.892536 
9.892432 
9.892334, 























9.801665 


10.100173}! 


Sine. 


Brae 
9.799028 





9.799651) 
9.799806 


9.799962 
9.8001 17 
9.800222) 
9.800477) 
9.800582 
9.800737 


9,800892 
9,801047 
9.801201 ' 
9.801356) 
9.801511 


9.801819 
9.801973}! 
9.802128 
9.802282 
9,802436 
9.802589 


9.802743 
9.802897 
9.803050! 
9.803204 
9.803357 
9.803514 


9.803664. 
9.803817 
9.803970 
9.804123! 
9.804276 
9.804428 


9.804581 
9.804734 
9.804886 
9.805039 
9.805191 
9,805 343 


9.805495 
9.805647 
9.805799 
9.805951 
9.806103 
9.806253 


9.806406 
9.806557 
9.806709 
9.806860, 
9.807111 

9.807165 


9.807314 
9.807465 











9.807615) 9.884572] 9.923034 10,076956 
9.807766) 9.884466} 9.923300)10.07 6700 


9.807917 | 
9. 9.808067; 


| Cosine. | 


i ses V8 DS EY AE SE SEES ae 
ee aeneashaniiantionanties’ 


te STiWW Ge SESE Nga Und oY. 
39 9 Deg. St ae 
Cosine. | 'T Tang. | Cotang. | 





9.890400) 9.908628) 10.091372159 
9. 799184) 9.890298) 9.908886) 10.091114158 
9.799839; 9.890195) 9.909 144) 10.090856 |57 
9 799495; 9.890093! 9.909402) 10.090598|56 


| 
9.890503] 9.908369 | 10.091631 a 


9.889990! 9.909660) 10.090340) 55: 
9.689888) 9.909918) 10.090082/54 


9.889785] 9 910177| 10.089823| 53: 
9.889682) 9.910435) 10.089565|52 


9.889579 
9.889477 
9.889374 
9.88927 1 


9.889168 
9.889064 
9.888961 
9.888858 
9.888755 
9.888651 


9.883548 
9.888444, 
9.888341 
9.888237 
9.888134 
9.888030 


9.887926 
9.887823 
9.887718 
9.887614 
9.87510 
9.887406 


9.887302 
9.887198 
9.887093 
9.886989 
9.886885 
9.886780 


9. 886676 
9.886571 
9.886466 
9.886362 
9.884257 
9.886152 


9.886047 
9.885942 
9.885837 
9.885732 
9.885627 
9.885522 


9.885416 
9.885311 
9.885205 
9.885100 
9.884994 
9.884889 


9.884783 
9.884677 


9.884360 


9.884254 


Sine. 





9.910693) 10.089307| 51 
9.910951) 10.089049| 50 
9.911209} 10.088791|49 
9.911467) 10,088533/48 


9.911725] 10.088275| 47 
9°911982) 10.088018]46 
9.912240] 1 0.087760) 45 
9,9 2498] 10.087502)44 — 
9,912756| 10.087 244/43 
9.913014|10.986986|42 


9,913271|10.086729|41 
9.913529] 10.08647 1 )40 
9.913787 |10.086213|39 
9.914044) 10.085956/38 
9.914303] 10,085698|37 
9,914560|10.085440|36 


9.914817) 10.085183] 35: 
9.915075} 10.084925|34 
9.915332) 10,084668/33 
9.915590) 10.084410]32 
9.915847] 10.084153] 31, 
9.916104] 10,083896| 30 


9.916362) 10.083638]29 
9,916619] 10.083381 |} 28 
9.916877) 10.083123}27 
9 917134) 10.082866}26 
9.917391 | 10.082609] 25 | 
9.917648) 10,082352) 24. 


9.917906) 10 0820941 23! 
9.918163] 10.081837|22 
9 918420) 10 081580) 21 
9.918677] 10.081323|20 
9 918934 10.081066 119] 
9,919191| 10.080809] 18. 


9.919448! 10.080552) 17 

9,919705{ 10.080295116 
9,919962} 10.080038) 15! 
9.920219] 10.079781 14) 
9,920476|10.079524113! 
9.920733} 10.079267 12) 


9.920996} 10.079710)11; 
9.921247 | 10.078753 10, 


9,921503|10.078497 
9.921760|10.078240 
9.922017 | 10.077983 
9.92297 4,|10.077726 


9.922530 }10.077470 
9.922784, 10.077213 


10,076443 
10.0761 86 


“Pang. 


9.925557 
9.923814 


~ Cotan._ 








Sine. 


\ Q, 
9.8085 19] 
. 9.808169: 
9.808819) 
9.808969 


9.809119 
9.809269 
9.809419 
9.809569 
9.809718 
9.809868 


9.810017 
9.810167 
9.810316 
9.810465 
9.810614 
9.810763) 


19 9.810912 
20) 9.811061! 
21] 9.811210) 
22) 9.811358 
23. 9.811507; 
4; 9.811655: 


SOeon onewnoeol - 





LOG) SINES, TANGENTS, ke: 


40 Deg. 


Cosine. 


9.808067] 9.884254 
| 9.808218! 9.884148 
808368! 9.883042 


9.883936 
9.883829 
9.883723 
9.883617 


9.883510 
9,88.3404 
9.883297 
9.883191 
9.883084 
9.882977 


9.882871 
9.882764, 
9.882657 
9.88255) 
9.882443 
9.882336 
9.882229 
9.882121 
9.882014 
9.881907 
9.881799 
9.881692 


‘9.926122 


Tang. 
9 923814, 
9.924070 
9.924.527 
9931583 
9.9 24840 
9.925096 
9.995.152 
9.925609 
9.925865 


Cotang. 


10,076186 
10.0075930 





10,.075417 
10.075160 
| 10.07 4904 
(10,074548 


10,07455: 
10,0741 35 
10.075878 
10.073622 
9,926634/ 10.073366 
9.926890) 10.073110 


9.927147) 10.072853) 
9.927403] 10.072597 
9.927659) 10,072341 
9.927915} 10.072085 
9.928171} 10.071829 
9.9 28427/10.071573 


9.928684! 10.071316 
9.98940] 10.071060 
9.939196] 10.070804 
9.929452'10.070548 
9,929708'10.070292 
9.929964 10,070036 


9.926378 








'40,075673) 


9.9302 -0|10.069780 
9,930475|10,069525 
9.93073 }110.069269 
9.930987 | 10.0690i3 
9.931243] 10,068757 
9.931499] (0.068501 


9,931755|10.068245 
9.95201 0] 10.067990 
9.932266) 10067734 
9,932522110.067478 
9.932778) 10.067222 
9.933033) 10,066967 


9.933289 10.0667 11 
9.933545 )10,.066455 
9.933800} 10.066200 
9.934056} 10.065944 
9.934311] 10.065689 
9.934567 | 10,065433 


9.934822) (0.065178 
9.935078] 10.064922 
9.935335} 10,064667 
9.935589|10.0644.11 
9.935 84.4.|10.064156 
9.936100} 10.063900 


ddaades| 10.063645 
9.936611 110.063389 
9.936866 | 10,063134 
9.937121] 10.062879 
9.937377 | 10.052623 
9.816069}. 9.878438} 9.937632] 10.062368 


9.816215; 9.878328) 9.937887) 10.062113 


9.881554 
9.881477 
9.881369 
9.881261 
9.881153 
9 881046 


9.880938 
9.880830 
9 880722 
9.880613 
9.880505 
9.880397 


9.880289 
9.880180 
9.880072 
9.879963 
9.879855 
9.879746 


9.879637 
9.879529 
.9,879420 
9.879311 
9.879202 
9.879093 


9.878984 
9.878875 
9.878766 
9.878656 


| 9.811804 
9.811952, 
, 9 812100 
» 9.812948) 

9.812396, 
9.819544 


9.812692 
| 9.812840 
9.812988) 
9.813533} 
9.813283) 
9.813430; 


9.813578 
| 9,81.3725 
9 813872 
9.814019 
9.814166 
9.814513 


9.814460 
9.814607 
9.814753 
9.814900 
9.815046 
9.815193 


9.815339 
9.815485 
9.815652 
9.815778 
9.815924) 9,878547 




















eS 
——— 





9.816361) 9.878219] 9.938% 42] 10:061858| 
9.816507 pha ad 9.938398110,061602 
| 





9.816652] 9.877999! 9.938653} 10,061347 
9.816798} 9.877890 9.938908) 10.061092 
9.816943) 9.877780 9.959163 


_ Sine. Cotan. 


49 Deg. 





Cosine. 


RTS 


| 


10 060837), 9.825511 


Tang. || Cosine. 


Sine Cosine. 


9.816943 | 9.877780] 9.939163 


9.817088! 9.877670 


9, 817233) 9 877560}. 


9.817379| 9.877450! 
9.817524) 9.877340 
9.817668; 9.877230 
9 817813] 9.877120 
9.817958! 9.877010 
9.818105! 9 376899 
9,818247| 9.876789 
9.818392) G 876578 
9.818536} 9.876568 
9.818681) 9.876457 


9,818825| 9.876347 
9.818969! 9.876236 
9.819113] 9.876125 
9.819257) 9.876014 
9.819401} 9.875904 
9.819545| 9.875793] 


9.819689} 9.875682 
9,819832| 9.875571 
9,819976) 9.875459 
9,820: 20) 9.875348. 
9.820263} 9.875237 | 
9.820 106| 9,875126 





4} Dee. 


Tang. | Cotang. 

|10.000837 |60 
9.939418 10,°6058215¢ 
9.939763!10.060527|58 
9.939998] 1 0.060072! 57 
9.940{ 53/10 059817|56 
9.940459/10.059561/ 55 
9.940694|10.05930615 4 


9.940949] 10.059051 | 53 
9.941204, 10.058796152 
9.94.1459|10.058541 | 51} 
recidichog tNapeiaetid 50 












9.941968! 10.058032149 
9,9.42223|10.057777) 48 


9,942478!10.057522147 
9.9427 33|10,057267 |46 
9,942988/10.057012/45 
9.943943110 056757 |44 
9.943498 10.056502)43 
9.9437 52|10.056248 


9.944007|10.055993 
9,944.262| 10.055738 
90,9445 17|10.055483 
9.94477 1}10.055229 
9,94,5026| 10.054974 
9.945281|10.054719 


rs 





“9.824808 
‘9.824949 


9.820550! 9.875014) 9.945535|10,054465 
9.820693| 9.874903). 9,945790| 10,054210 
9.820836] 9.874791) 9.946044 10,053955 
9.820979) 9.874680| 9.946299/10.053701 
9.821122 9.874568) 9.946554) 10,053446 
9.821265| 9.874456) 9.946808|10.053192 


9.821407! 9.874344). 9.947063|10.052937 
9.821550) 9.874232! 9.947318/10.052682 
9.821693] 9.874121 9.947572|10.052428 
9.821835) 9.874009; 9.947827|10.052173 
9.821977{ 9.873896) 9.948081 |10.051919 
9.822120! 9.873784, 9.948335|10.051665 


9,822969| 9.873672) 9.948590|10.051410 
9.822404 9.873560] 9.948944| 10.051756 
9,829546| 9.873448] 9.949099|10.050901 
9.822688! 9.873335} 9.949353110.050647 
9.829830) 9.873223} 9,949608/10.050392 
9.822972] 9.873110] 9,949862|10.050158 


9.823141 9.872998! 9.950116|10,049884) 17 
9.823955| 9.872885] 9.950371/10.049629| 16 
550 9.872772 10.04.9375] 15 

















9.950625 
9.823539] 9.872659! 9.950879110.049121 | 14 
9,823680| 9:872547). 9.951133} 10.048867|13 
9.823821] 9.872434 


9.951388!10.048612| 12 
9.823963) 9.872321 


9.951642) 10,048358] 11] 
9.824104| 9 872208} 9.951896) 10.0481 04/10 
9.824245) 9.872095} 9.952150}10.047850 
9.824.386! 9.871981! 9.952405] 10.047595 
9,824597| 9.871868) 9.952659110.047341 
9.824668) 9.871755] 9.952913/10.047087| © 
9.871641 


9.953167 |10.046833 
9.871528) 9.953421 |10.046579! 
9.871414) 9.953675 |10.046325 
9 871301) 9.953929|10.046071 
9.871187} 9.954183!10.045817 
9.871073, 9.954437|10.045563 


Sine. Gotan. _| Tang. 
48 Deg. 








9.825230 


0.805230 
9.825371 

















oom 





- 


LOG. SINES, TANGENTS, &c. 


42 Deg. ‘ 
Sine. Cosine. Tang. _Cotang. 
9.825511! 9.871073] 9.954437) 10.045563 
9.825651} 9.870960) 9.954691/10,045309 
osasta 9.870846) 9.954946/10.045054 


Sine. Cosine. 
9.833783 
9.833919 
9.834054 
9.834189 
9.834325 
9.834460 
9°834595 
9.834750 
9.834865 














9.364010) 
9.863892 
9.863774 
9.863656 
9.863538 
9.863419 


9.853301 
9.863183 
9.834999] 9.863064 
| 9.835134) 9.869946 
| 9.835269| 9.862827, 

9.835403) 9.862709 
9 835538) 9.862590 
9.835672! 9.862471! 
9.835807| 9.862353) 
9.8 
9; 


9.825931} 9.870732) 9.955200) 10.044800 
9.826071; 9.870618 


9.955454 10.044546) 
9.826211] 9.870504 


9.955708|10,044292 
9.826351) 9.870390| 9 955961|10.042039| 


7| 9.826491| 9.870276, 9.956215|10.0437 85 
8| 9.826631) 9.87016!| 9.956469) 10.043531 
i 9.826770| 9.870047| 9.956723] 10.043277|| 


> ib Got = © 











10} 9.826910} 9.869933) 9.956977|10.043023 
11} 9.827049] 9.869818] 9.957231} 10.042769 
9.957485) 10,042515 





9 827189] 9.869704 


12 

is 9.827329 9.869589) 9.957739 10.042261| 
14) 9.827467! 9.869474; 9.957993 10,042007 | 
15 9.827606) 9.869360, 9.958247 10.041753| 
16} 9.827745: 9.869245! 9.958500 10.041500) 
9.827884: 9.869130] 9,958754110 041246 
9.828023] 9.869015} 9.959008} 10.040992) 


9.828162} 9.868909} 9.959262) 10.040738 
20} 9.828301| 9.868785 
21' 9.828439! 9.868570 
22) 9.828578] 9.868555 
23) 9,828716| 9.868440 








35941 | 9.862234, 
836075 
9.836209 


9.836343 


17 
18 


9.862115) 
9.861996 


9,861 877 
9.959516! 10.040484|| 9.836477] 9.861758: 
9,959769| 10.040231|! 9.836611! 9.861638 
9,960023|10.039977;| 9,836745' 9.861519 
9.960277 |10.039723/| 9.836878! 9.861400, 
« 9.828855 9.868324 9.860530] 10,039470)) 9.857012) 9.861280 


125! 9.828993] 9.868209! 9.960784110.039216 9.837146] 9 861161 
26; 9.829131 9.868093 9,961038/10.038962|| 9.837279) 9.861041) 
_}27} 9.829969} 9.867978) 9.961292! 10038708]! 9.887412) 9.860922 
28) 9.829407} 9.867862! 9.961545) 10.038455}| 9.837546 9-860802) 
29) 9,829545| 9.867747) 9.961799/10.03x201 ossrst2 9.860682, 























30] 9.829683} 9.867631} 9.962052) 10,037948|| 9.837812} 9.860562 
S1| 9.829821 tr 





9.962306|10.037694 
9.962560|10.037440 
9.962813] 10,037 187 
9.963067|10.036933 
9.963320|10.036680) 
9.963574) 10,036426 


9.963828] 10.036172 
9.964081 | 10.035919 
9.964335] 10.035665 
9.964588) 10.035412 
9.964842|10.035158 9.839272| 9.859239! 
9.965095|10.034905|! 9.839404; 9.859119 


9.965349] 10.034651|! 9.839536] 9.858998 
9.965602 10.034598| 0.839668} 9.858877) 
9.965855|10.034145|| 9.839800! 9.858756 
9.966109] 10.033891|| 9.839932| 9.858635 
9.966362] 10.033638)| 9.840064| 9.858514 
9.966616|10.033384|| 9.340196] 9.858393 


9.966869] 10.033131|| 9.840328] 9.858972 
9.967 123] 10.032877|| 9.840459} 9.858151 
51| 9.832561} 9.865185] 9.967376) 10.032624,| 9.840591) 9.858029 
52) 9.832697] 9.865068] 9,967629|10.032371|! 9.840722] 9.857908 
z 9.832833] 9.864950! 9.967885|10.032117|| 9.840854! 9.857786 


9,837945| 9.860442 
9.838078} 9.860322. 
9,838211} 9.860202 
9.838344) 9.860082 
9,838477| 9.859962 
9.838610} 9.859842 


9,838742| 9.859721 
9.838875) 9.859601 
9.839007| 9.859480 
9.839140} 9.859360: 





32| 9.829959] 9.867399 
9.830097| 9.867283 
9.830234| 9.867167 
9.830372] 9.867051 
9.830509] 9.866935 


9.830646 25609 
9.830784| 9.866703 
9,830921| 9.866586 
9.831058} 9.866470 
9.831195] 9.866353 
42| 9.831332] 9.866237 


43) 9.831469) 9.866120 
44) 9.831606] 9.866004 
45| 9.831742] 9 865887 
46| 9.831879] 9.865770 
47| 9.832015] 9.865653 
48) 9.852152] 9.865536 


49] 9.832288] 9.865419 
50) 9.832425 seta 





34 
35 
36 


37 
38 
39 
40 
41 


: 








| 









































54| 9.832969) 9.864833] 9.968136) 10.031864)| 9.840985) 9.857665) 


55! 9.833105] 9.864716: 9.968389 10.031611| 
56] 9.833241] 9.864598 9.968643] 10.031257 | 
57| 9.833377| 9°864481| 9.968896'10.031104 
58} 9.833512! 9.864363! 9.969149!10.030851|| 9.841509 
59] 9.833648| 9.864245! 9.969403! (0.030597/| 9.841640 
60| 9.833783] 9.864127} 9.969656/10 030344!;-9.841771 


~~ | Cosine. Sine. | Cotan. — “Tang. 1 
47 Deg. 


9.841116 
9.841247 
| 9.841378 


9.857543 
9.857422 
9.857300 
9.857178 
9.857056 
| 9.856934 














Cosine. 














43 Deg.. 





9.864127} 9.969656 10,030344) 60 


Sine. | Cotan, | Tang. 
46 Deg. { 





Tang. | Cotang. | 

9.969909] 10.030091 59 
9.970162] 10,029838|58 
9.970416] 10,029584157 
9,970669| 10,029331156 
9.970929} 10,029078)55} 
9.971175) 10,028825|54 


9.971429] 10.098571153 
9,97 1682) 10.028318|52 
9.971935|10.028065151 
9.972188] 10.027812|50 
0.972441 |10.027559149 
9.979695] 10.027305 148 


9,97 2948} 10.027052147 
9.97320) |10,026799|46 
9.973454] 0.026546 /45 
9.973707] 10.026293 44 
9.973960)10.026040'43 
9,9742 3) 10,0257 87 42 


9.974466 10,025584/41 
9.9747 2|10.025280, 40} 
9.97497 3| (0.025027 | 39 
9.973226|1 0.024774 38 
9.975 .79|10,024521|37 
9.975732! 10,024-268|36 | 


9.975985|10.024015135} 
9.97 6238 | 10,023762|34 
9 976491 |10.023509|33 
9 976744] 10,023256!32} 
9.976997] 10,023003' 3k 
9.977 250|10,022756|30 


9,97750.3|10.022497 |29 } 
9.977756) 1002224428! 
9,97 8009|10.021991 | 27 | 
9.978262) 10.021738 26} 
9,978515|10,021 48525 | 
9.978768) 10.021 235 24} 


9.979021 | 10,020979|23| 
9.979274] 10.020726|22 | 
9,979527|10.020473'21} 
9,979780}|10.020220|20) 
9.980033) 10,019967 |19} 
9,980286/15,019714/18 


9.980535] 10.019462) 17 | 
9.980791 
9,981044/10. 
9.981297) 10.0; 8703114} 
9.981550) 10.018450)13 
9.981803 10.018197)12 


9.982056|10.017944| 11 | 
9.982309 10.017691) 103 
9.982562) 10.017438! 9 
9.982814/10.017186| 8) 
0,983067 10.016933| 7 
9.983320|10.01 6680) 6 


9.983573) 10.016427| 5) 
9.983826) 10.01.6174 
9.984079}10.03 5921) 3 
9,984532'10.015668) 2 
9.9845 84'10.015416 : 

0 






















be 


9,984837 |10.015163 





senna aero ahd 


LOG. SINES, TANGENTS, &c. a 





! Sine. Cosine. 
i 9.841771 9.856934 
1} 9.841902] 9.856812 
2! 9.842033} 9.856690 
3} 9.842163) 9.856568 
4| 9,842294| 9.856446 
9.842424} 9.856323 
9.842555} 9.856201 


6 

A 9.842685] 9.856078 
8] 9.842815] 9.855956) 
G| 9.842946]. 9.855833 
ti 9.843076} 9.855711 
; 9.843206] 9.855588 


9.845336] 9.855465 


3! 9.843466} 9.855342 
4] 9.843595 9.852519 
9.843725} 9.855096 
Petes 9.854973 
u7| 9,843984| 9.854850 
18 9.844114) 9.854727 


Ia 9.844243] 9.854603 
120] 9.844372) 9.854480 
j2L| 9.844502] 9.854356 
\22) 9.844631] 9.854233 
9.844760} 9.854109 


2 
: 9.844889] 9.858986 
25 


9.845018] 9.853862 
9.845147] 9.853738 
9.845276} 9.853614 
al 9: 845405| 9.853490 
9.845533] 9.853366 
a ogasens 9.853242 


31| 9.845790) 9.855118 
32} 9.845919] 9.852994. 
4,846047| 9.852869 
9.846175} 9.852745 
35! 9.846304, 9.852620 
36] 9.846432) 9.852496 


437} 9.846560} 9.852371 
38} 9.846688} 9.852247 
39} 9.846816] 9.852122 
40! 9.846944] 9.851997 
41) 9.847071) 9.851872 
42) 9,8471991 9.851747 








43} 9.847327) 9.851622 


44| 9.337454| 9.831497 
45! 9.847582] 9.851372 
46| 9.847709! 9.851246 
47| 9.847836) 9.851191 
148] 98479641 9.851996 


49} 9.848091} 9.850870 
. (50) 9.848218) 9.850744 
51} 9.848345} 9.850619 
52] 9.848472) 9.850493 
53| 9.848599} 9.850368 
54| 9.848726] 9.850242] 9 


55| 9.848852) 9.850116 
pa, 9.848979} 9.849990 

7! 9 849106) 9.849864 
8 9.849232) 9.849738 
| 9.849359} 9.849611 


| Cosine. Sine. 













44 Deg. 


60 9,849485] 9.849485] 10.000000)10,000000 

~ Cotan. “Tang. |’ 

“45 Deg. | 
FINES. 


eee 


| Tang. . | Cotang. | 


9.984837|10.015 163]60 
9,985090]10.014910159 
9.985343|10,014657|58 
9.985596) 10.014404|57 
9.985848] 10,014152156 
9,986101]10,013899155 





9.986354|10.013646|54 


9,986607|10,013393153 
9.986860}10,01 3140152 
9,987 112|10,012888)51 
9.987365|10,012635|50 
9.98761 8|10.012382)49 
9.987871) 10,012129|48 


9,988 123/10.011877|47 
9.988376) 10.011624)46 
9.988629} (0.01137 1/45 
9.988882/10.011118]44 
9.9891 34.|10.010866)43 
9.989387 |10.010613/42 


9.989640! 10.010360)41 
9.989893) 10.010107)40 
9.990145) 10.009855|39 
9.990398) 19.009602/38 
9.990651 |10,009549|37 
9.990903) 10,009097/36 


9.991156] 10.008844|35 
9.99 § 4.09] 10.008591/34 
9.991662) 10,008338|33 
9.991914! 10,008086/32 
9.992167|10,007833/31 
9.992420) 10,007580)30 


9.992672) 10,007328/29 
9.992925! 10.007075|28 
9.993178) 10.006822|27 
9.993431|10,006569/26 
9.993683) 10,006317/25 
9.993936 |10.006064\24 


9.994189 }10.005811/23 
9.99444.1}10,005559|22 
9.994694]10,005306/2L 
9.994.947 110,005053/20 
9.995199/10,004801)19 
9.995452] 10,004.548|18 


9.995705 |10.004.295|17 
9,995957|10.00404.3}16 
9,996210]10.003790) 15 
9.996463|10.003537|14 
9.996715} 10.003285|13 
9.996968/10.003032)12 


9.997223 |10.002779) 01 
9.997473] 10.002527|10 
9.997726] 10.002274 
9.997979] 10,002021 
9.998231) 10,001769 

9.998484 10.001516 


9,9987'37|10.001263 

9.998989110.001011 
9.999242|10.000758 
9.999495 |10.000505 
9.999747 |110.000253 








S=ewOkn ANWOO 


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URBANA 


UNIVERSITY OF ILLINOIS- 


3 0112 018279528 


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Peet ee 
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